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            Lecture 17: Counting Rules II
         
    
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<p><strong>Description:</strong> Covers computing cardinality of sets with inclusion–exclusion, the bookkeeper rule, the subset rule, and poker hands with applications to probability and counting.</p> <p><strong>Speaker:</strong> Marten van Dijk</p>
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<p><span m="390">The</span> <span m="520">following</span> <span m="950">content</span> <span m="1550">is</span> <span m="1660">provided</span> <span m="2110">under</span> <span m="2390">a</span> <span m="2430">Creative</span> <span m="2830">Commons</span> <span m="3230">license.</span> <span m="4350">Your</span> <span m="4530">support</span> <span m="5030">will</span> <span m="5200">help</span> <span m="5440">MIT</span> <span m="5890">OpenCourseWare</span> <span m="6680">continue</span> <span m="7190">to</span> <span m="7270">offer</span> <span m="7690">high</span> <span m="7920">quality</span> <span m="8450">educational</span> <span m="9070">resources</span> <span m="9700">for</span> <span m="9850">free.</span> <span m="11050">To</span> <span m="11060">make</span> <span m="11260">a</span> <span m="11310">donation</span> <span m="12000">or</span> <span m="12260">view</span> <span m="12700">additional</span> <span m="13130">materials</span> <span m="13670">from</span> <span m="13810">hundreds</span> <span m="14250">of</span> <span m="14360">MIT</span> <span m="14780">courses,</span> <span m="15900">visit</span> <span m="16120">MIT</span> <span m="16530">OpenCourseWare</span> <span m="17580">at</span> <span m="17750">ocw.mit.edu.</span> </p>
<p><span m="24730">PROFESSOR: We're</span> <span m="24870">going</span> <span m="25090">to</span> <span m="25210">talk</span> <span m="25490">about</span> <span m="26250">inclusion-exclusion</span> <span m="28650">which</span> <span m="28910">is</span> <span m="29370">a</span> <span m="29460">generalization</span> <span m="30400">of</span> <span m="30880">what</span> <span m="31090">we</span> <span m="31200">did</span> <span m="31800">at</span> <span m="31960">the</span> <span m="32070">end</span> <span m="32250">of</span> <span m="32400">last</span> <span m="32950">lecture,</span> <span m="35230">which</span> <span m="35350">was</span> <span m="35540">about</span> <span m="35790">the</span> <span m="35850">sum</span> <span m="36820">rule.</span> <span m="41550">Everybody's</span> <span m="42070">still</span> <span m="42330">talking</span> <span m="42750">I</span> <span m="42810">hear.</span> <span m="43890">Anyway,</span> <span m="44440">the</span> <span m="44830">inclusion-exclusion</span> <span m="46200">principle</span> <span m="46890">is</span> <span m="47120">very</span> <span m="47320">important.</span> <span m="48290">And</span> <span m="49280">the</span> <span m="49380">best</span> <span m="49660">way</span> <span m="49770">to</span> <span m="49890">explain</span> <span m="50330">this</span> <span m="50790">is</span> <span m="51010">by</span> <span m="51150">using</span> <span m="52951">Venn</span> <span m="53340">diagrams.</span> <span m="54680">So</span> <span m="54840">let's</span> <span m="55060">do</span> <span m="55190">this.</span> </p>
<p><span m="56540">So</span> <span m="56740">what</span> <span m="57010">did we</span> <span m="57120">do</span> <span m="57250">before?</span> <span m="57890">When</span> <span m="58110">we</span> <span m="58540">worked</span> <span m="58850">on</span> <span m="58950">a</span> <span m="59030">sum</span> <span m="59300">rule,</span> <span m="59890">we</span> <span m="60240">saw</span> <span m="60620">that</span> <span m="60940">if</span> <span m="61160">we</span> <span m="61310">count</span> <span m="62590">the</span> <span m="62720">union</span> <span m="63280">of</span> <span m="64720">a</span> <span m="64819">whole</span> <span m="65800">collection</span> <span m="66340">of sets</span> <span m="66840">that</span> <span m="66940">are</span> <span m="67120">all disjoints,</span> <span m="68070">then</span> <span m="68200">we</span> <span m="68290">can</span> <span m="68450">just</span> <span m="68780">some</span> <span m="69150">all the</span> <span m="69440">cardinalities</span> <span m="70170">of</span> <span m="70330">each</span> <span m="70590">of</span> <span m="70760">the</span> <span m="70810">members.</span> <span m="72290">So</span> <span m="73770">in</span> <span m="73910">the</span> <span m="74030">case</span> <span m="74390">where we</span> <span m="74720">have</span> <span m="75160">intersections--</span> <span m="76020">so</span> <span m="76530">sets</span> <span m="76850">are</span> <span m="76910">not</span> <span m="77110">disjoint--</span> <span m="78030">well,</span> <span m="78280">if</span> <span m="78420">you</span> <span m="78490">just</span> <span m="78990">count</span> <span m="79560">all</span> <span m="79740">the</span> <span m="79820">cardinalities--</span> <span m="80440">all of</span> <span m="80640">the</span> <span m="80700">sizes</span> <span m="81150">of</span> <span m="81250">the</span> <span m="81350">sets</span> <span m="81640">together--</span> <span m="82500">you'll</span> <span m="82700">start</span> <span m="82980">to</span> <span m="83040">do</span> <span m="83150">some</span> <span m="83320">double</span> <span m="83580">counting.</span> </p>
<p><span m="85190">So</span> <span m="85420">with</span> <span m="85760">the</span> <span m="86030">inclusion-exclusion</span> <span m="86980">principle,</span> <span m="87600">we</span> <span m="87620">can</span> <span m="88170">actually</span> <span m="90880">exactly</span> <span m="91460">compute</span> <span m="92350">the</span> <span m="92480">cardinality</span> <span m="93130">of</span> <span m="93260">the</span> <span m="93380">union</span> <span m="93740">of</span> <span m="93820">sets.</span> <span m="97910">And</span> <span m="98120">you</span> <span m="98220">will</span> <span m="98330">see</span> <span m="98570">that</span> <span m="98920">this</span> <span m="99200">principle</span> <span m="99850">can</span> <span m="100020">be</span> <span m="100100">used</span> <span m="100870">in</span> <span m="101260">plenty</span> <span m="101535">of</span> <span m="101810">examples,</span> <span m="102710">including</span> <span m="103950">some</span> <span m="104230">of</span> <span m="104380">them</span> <span m="104700">that</span> <span m="105010">are</span> <span m="105330">on</span> <span m="105410">the</span> <span m="105470">problem</span> <span m="105750">set.</span> <span m="109740">So</span> <span m="110040">how</span> <span m="110140">does</span> <span m="110310">it</span> <span m="110440">work?</span> </p>
<p><span m="111740">So</span> <span m="111960">let's</span> <span m="112330">set</span> <span m="112580">take</span> <span m="112790">an</span> <span m="112850">example</span> <span m="113440">where</span> <span m="113590">we</span> <span m="113730">just</span> <span m="113990">have</span> <span m="114120">two</span> <span m="114330">sets</span> <span m="114950">and they</span> <span m="115130">have</span> <span m="115310">an</span> <span m="115380">intersection.</span> <span m="117050">So</span> <span m="117210">say</span> <span m="117440">this</span> <span m="117670">is</span> <span m="117780">one</span> <span m="118030">set</span> <span m="119210">which</span> <span m="119400">we</span> <span m="119500">call</span> <span m="119790">M,</span> <span m="120790">and</span> <span m="121130">say</span> <span m="121360">here is</span> <span m="121740">another</span> <span m="122050">set</span> <span m="123420">which</span> <span m="123610">we</span> <span m="123690">call</span> <span m="124030">E.</span> <span m="126640">Well,</span> <span m="127340">they</span> <span m="127450">have an</span> <span m="127830">intersection--</span> <span m="128370">there's</span> <span m="128550">some</span> <span m="128720">overlap</span> <span m="129229">over</span> <span m="129500">here.</span> <span m="130530">So</span> <span m="130900">if</span> <span m="131080">you</span> <span m="131250">would</span> <span m="131480">like</span> <span m="131760">to</span> <span m="132050">write</span> <span m="132310">out</span> <span m="132650">the</span> <span m="132750">different</span> <span m="134000">areas</span> <span m="134600">over</span> <span m="134840">here,</span> <span m="135160">what</span> <span m="135720">they</span> <span m="135890">were equal</span> <span m="136220">to,</span> <span m="136420">well,</span> <span m="137050">this</span> <span m="137280">is</span> <span m="137950">everything</span> <span m="138520">that</span> <span m="138610">is in</span> <span m="138690">M,</span> <span m="139870">but</span> <span m="140100">we</span> <span m="140300">exclude</span> <span m="141870">what</span> <span m="142040">is in</span> <span m="142370">E.</span> <span m="142830">So</span> <span m="143510">you</span> <span m="143990">write</span> <span m="144290">it</span> <span m="145820">M</span> <span m="146580">backslash</span> <span m="147420">E.</span> </p>
<p><span m="149610">This</span> <span m="149920">part</span> <span m="150190">over</span> <span m="150450">here</span> <span m="150800">is</span> <span m="151050">the</span> <span m="151990">intersection</span> <span m="152940">of--</span> <span m="153560">this</span> <span m="153740">maybe</span> <span m="154170">is</span> <span m="154250">a little bit too</span> <span m="154370">small--</span> <span m="155490">so</span> <span m="155630">this</span> <span m="155810">is</span> <span m="155910">the</span> <span m="156010">intersection</span> <span m="156700">of</span> <span m="156810">both</span> <span m="157070">sets.</span> <span m="162320">And</span> <span m="162520">over</span> <span m="162740">here</span> <span m="163000">we</span> <span m="163120">have</span> <span m="164050">everything</span> <span m="164490">that's</span> <span m="164710">in</span> <span m="164870">E</span> <span m="165330">but</span> <span m="165630">excluding</span> <span m="166170">what</span> <span m="166330">is</span> <span m="166570">in M.</span> <span m="167930">So</span> <span m="168110">if</span> <span m="168230">you</span> <span m="168310">just</span> <span m="168590">write</span> <span m="168790">out</span> <span m="169330">the</span> <span m="169470">cardinalities</span> <span m="170150">of the</span> <span m="170450">different</span> <span m="170880">sets that</span> <span m="171040">we're</span> <span m="171340">interested</span> <span m="171940">in--</span> <span m="172220">so</span> <span m="172270">we</span> <span m="172340">would</span> <span m="172500">like</span> <span m="172680">to</span> <span m="172790">compute</span> <span m="173260">the</span> <span m="173350">cardinality</span> <span m="173950">of</span> <span m="174080">the</span> <span m="174200">union</span> <span m="174580">of</span> <span m="174740">M</span> <span m="175280">and</span> <span m="175500">E.</span> </p>
<p><span m="176680">So</span> <span m="176870">let's</span> <span m="177760">have</span> <span m="177910">a</span> <span m="178000">look</span> <span m="178380">how</span> <span m="180520">the</span> <span m="180630">cardinality</span> <span m="181200">of</span> <span m="181350">M</span> <span m="181490">itself</span> <span m="181850">can</span> <span m="182000">be</span> <span m="182130">computed.</span> <span m="182590">Well,</span> <span m="183740">if</span> <span m="183900">you</span> <span m="183980">use</span> <span m="184250">the</span> <span m="184510">sum</span> <span m="184850">rule</span> <span m="185310">which</span> <span m="185460">says</span> <span m="185890">that</span> <span m="186070">we</span> <span m="186250">can</span> <span m="187090">simply</span> <span m="188890">add</span> <span m="191480">the</span> <span m="191630">two</span> <span m="191870">disjoint</span> <span m="192450">sets</span> <span m="192810">that</span> <span m="192930">make up</span> <span m="193360">set</span> <span m="193640">M--</span> <span m="194120">we</span> <span m="194240">can</span> <span m="194570">count</span> <span m="194880">each</span> <span m="195130">of</span> <span m="195290">those</span> <span m="195870">cardinality--</span> <span m="196560">so</span> <span m="196780">we</span> <span m="196890">have</span> <span m="197910">M</span> <span m="198220">consisting</span> <span m="198910">of</span> <span m="199130">this</span> <span m="199400">part,</span> <span m="199960">which</span> <span m="200200">is</span> <span m="200460">M</span> <span m="200680">excluding</span> <span m="201160">E,</span> <span m="201510">plus</span> <span m="202590">the</span> <span m="202710">part</span> <span m="203000">that</span> <span m="203110">is</span> <span m="203280">right</span> <span m="203500">here</span> <span m="204410">which</span> <span m="204590">is</span> <span m="204700">the</span> <span m="204780">intersection.</span> <span m="205550">So</span> <span m="206110">we</span> <span m="206210">use</span> <span m="206420">the</span> <span m="206470">sum</span> <span m="206710">rule</span> <span m="208790">and</span> <span m="209200">conclude</span> <span m="209900">that</span> <span m="211660">we</span> <span m="212030">have</span> <span m="215040">the</span> <span m="215150">sum</span> <span m="215620">of</span> <span m="216010">everything</span> <span m="216440">that's</span> <span m="216670">in</span> <span m="216790">M</span> <span m="216980">excluding</span> <span m="217540">E</span> <span m="217860">plus</span> <span m="218990">the</span> <span m="219140">cardinality</span> <span m="219980">of</span> <span m="220220">the</span> <span m="220290">intersection.</span> </p>
<p><span m="222020">Now,</span> <span m="222150">for</span> <span m="222310">set E</span> <span m="222690">we</span> <span m="222770">can</span> <span m="222940">do</span> <span m="223050">something</span> <span m="223450">completely</span> <span m="223870">similar.</span> <span m="225560">So</span> <span m="225860">we</span> <span m="226000">will</span> <span m="226170">have</span> <span m="227290">everything</span> <span m="227720">that's</span> <span m="227920">in</span> <span m="228080">E</span> <span m="228350">excluding</span> <span m="228970">what</span> <span m="229130">is</span> <span m="229290">in</span> <span m="229800">M.</span> <span m="231108">And</span> <span m="231980">then</span> <span m="232190">we</span> <span m="232290">still</span> <span m="232630">have</span> <span m="233360">the</span> <span m="233440">intersection.</span> <span m="235540">Now</span> <span m="235730">we</span> <span m="235910">are</span> <span m="236180">interested</span> <span m="237830">in</span> <span m="238710">the</span> <span m="238890">union</span> <span m="239570">of</span> <span m="239670">the</span> <span m="240015">sent</span> <span m="240360">M</span> <span m="240610">and</span> <span m="240890">E.</span> <span m="241810">And</span> <span m="242850">now</span> <span m="243030">we</span> <span m="243120">need</span> <span m="243310">to</span> <span m="243650">add</span> <span m="244330">this</span> <span m="244650">one</span> <span m="244940">this</span> <span m="245060">one</span> <span m="245300">and</span> <span m="245400">this</span> <span m="245440">once</span> <span m="245630">together.</span> <span m="246220">So</span> <span m="246520">what</span> <span m="246670">do</span> <span m="246760">we</span> <span m="246890">see?</span> <span m="247280">We</span> <span m="247380">see</span> <span m="247510">that</span> <span m="247670">this</span> <span m="247850">is</span> <span m="247970">actually</span> <span m="248290">equal</span> <span m="249250">to</span> <span m="249470">everything</span> <span m="249970">that</span> <span m="250050">is in</span> <span m="250390">M</span> <span m="250690">but</span> <span m="250880">not</span> <span m="251110">in</span> <span m="251270">E,</span> <span m="252700">everything</span> <span m="253470">that</span> <span m="253630">is</span> <span m="253790">in</span> <span m="253910">their</span> <span m="254090">intersection,</span> <span m="256339">plus</span> <span m="257029">everything</span> <span m="257294">that's</span> <span m="257769">in</span> <span m="257940">E</span> <span m="258350">but</span> <span m="258500">not</span> <span m="258850">in</span> <span m="258990">M.</span> </p>
<p><span m="260810">So</span> <span m="261040">we</span> <span m="261140">have</span> <span m="261339">these</span> <span m="263300">equations</span> <span m="264010">that we</span> <span m="264100">can</span> <span m="264340">directly</span> <span m="264730">derived</span> <span m="265110">from</span> <span m="265410">the</span> <span m="265660">sum</span> <span m="266250">rule.</span> <span m="267150">And,</span> <span m="268780">well,</span> <span m="269660">now</span> <span m="269830">we</span> <span m="269960">can</span> <span m="270210">see</span> <span m="270560">here</span> <span m="271060">that</span> <span m="271340">if</span> <span m="271640">we</span> <span m="271900">are</span> <span m="272060">going</span> <span m="272440">to</span> <span m="272870">simply</span> <span m="273490">add--</span> <span m="274770">that's</span> <span m="275000">what</span> <span m="275120">the</span> <span m="275200">sum</span> <span m="275400">rule</span> <span m="275610">would</span> <span m="275750">say--</span> <span m="276330">simply</span> <span m="276730">add</span> <span m="276840">the</span> <span m="276910">cardinality</span> <span m="278160">of</span> <span m="278340">M</span> <span m="278550">and</span> <span m="278950">of</span> <span m="279230">E,</span> <span m="280340">well,</span> <span m="280510">if</span> <span m="280660">we</span> <span m="280790">add</span> <span m="281120">those</span> <span m="281370">two</span> <span m="281530">together,</span> <span m="282520">we will</span> <span m="282790">count</span> <span m="283180">the</span> <span m="283270">intersection</span> <span m="283930">double.</span> <span m="285240">So</span> <span m="285490">we'll</span> <span m="285580">need</span> <span m="285730">to</span> <span m="285830">cancel</span> <span m="286240">that</span> <span m="286420">because</span> <span m="286710">it's</span> <span m="286870">only</span> <span m="286980">counted</span> <span m="287440">once</span> <span m="288680">in</span> <span m="288840">the</span> <span m="288930">union.</span> </p>
<p><span m="290780">So</span> <span m="292050">if</span> <span m="292330">we</span> <span m="293200">combine</span> <span m="293620">those</span> <span m="293950">three</span> <span m="294760">over</span> <span m="295070">here--</span> <span m="295880">those</span> <span m="296120">three</span> <span m="296400">equations--</span> <span m="297970">then</span> <span m="300020">we</span> <span m="300130">can</span> <span m="300300">conclude</span> <span m="300870">that</span> <span m="301640">this</span> <span m="302890">counts</span> <span m="305840">the</span> <span m="305940">set</span> <span m="306850">M</span> <span m="307140">intersecting</span> <span m="307900">with</span> <span m="308160">E</span> <span m="308510">twice.</span> <span m="310320">So</span> <span m="311020">we</span> <span m="311270">have</span> <span m="311640">that</span> <span m="313210">the</span> <span m="313370">union</span> <span m="314180">is</span> <span m="314380">really</span> <span m="314770">equal</span> <span m="315750">to</span> <span m="316630">the</span> <span m="316750">cardinality</span> <span m="317450">of</span> <span m="317610">M</span> <span m="317950">plus</span> <span m="318330">that</span> <span m="318980">of</span> <span m="319250">E</span> <span m="320400">minus</span> <span m="320950">their</span> <span m="321130">intersection.</span> <span m="323280">Now</span> <span m="323410">we</span> <span m="323500">can</span> <span m="323690">generalize</span> <span m="324300">this.</span> <span m="325650">And</span> <span m="325890">we</span> <span m="325940">will</span> <span m="326060">give</span> <span m="326370">one</span> <span m="326520">more</span> <span m="326700">example</span> <span m="327460">where</span> <span m="327600">we</span> <span m="327710">have</span> <span m="327880">three</span> <span m="328130">sets</span> <span m="331240">and</span> <span m="331380">it</span> <span m="331470">will</span> <span m="331720">give</span> <span m="331910">the</span> <span m="332400">general</span> <span m="333200">equation.</span> </p>
<p><span m="334240">So</span> <span m="335670">suppose</span> <span m="336070">you</span> <span m="336200">have</span> <span m="336410">three</span> <span m="336620">sets</span> <span m="336960">now.</span> <span m="337780">Like</span> <span m="338020">one,</span> <span m="339290">another</span> <span m="339600">one,</span> <span m="341660">and</span> <span m="341750">another</span> <span m="342020">one</span> <span m="342210">over</span> <span m="342470">here.</span> <span m="342840">So</span> <span m="342930">we</span> <span m="343060">have</span> <span m="343330">M,</span> <span m="344635">E,</span> <span m="345940">and,</span> <span m="346140">say,</span> <span m="346390">the</span> <span m="346490">set</span> <span m="346830">S.</span> <span m="348430">Well,</span> <span m="348790">let's</span> <span m="348980">have a</span> <span m="349070">look</span> <span m="349480">how</span> <span m="349630">many</span> <span m="349960">times</span> <span m="351510">these</span> <span m="351800">different</span> <span m="352300">areas--</span> <span m="352880">they're</span> <span m="353040">all</span> <span m="353640">disjoint--</span> <span m="355530">how</span> <span m="355680">many</span> <span m="355920">times</span> <span m="356190">they</span> <span m="356380">are</span> <span m="357070">counted</span> <span m="358080">when</span> <span m="358280">me</span> <span m="358910">add</span> <span m="359860">the</span> <span m="360090">three</span> <span m="362620">cardinalities</span> <span m="363570">together.</span> </p>
<p><span m="364100">So</span> <span m="364280">we</span> <span m="364390">have</span> <span m="364910">M</span> <span m="365550">plus</span> <span m="367620">the</span> <span m="367710">cardinality</span> <span m="368350">of</span> <span m="368470">E</span> <span m="368760">plus</span> <span m="369060">the</span> <span m="369160">cardinality</span> <span m="369820">of</span> <span m="369990">S.</span> <span m="371210">Well,</span> <span m="371360">if</span> <span m="371510">you</span> <span m="371600">do</span> <span m="371760">that</span> <span m="372470">then</span> <span m="372610">we</span> <span m="372740">count</span> <span m="373240">this</span> <span m="373440">area</span> <span m="373910">exactly</span> <span m="374430">once</span> <span m="374880">because</span> <span m="375230">it's</span> <span m="375410">in</span> <span m="375570">M,</span> <span m="376380">this</span> <span m="376580">area</span> <span m="376990">we</span> <span m="377070">count</span> <span m="378150">once</span> <span m="378520">because</span> <span m="378870">it's</span> <span m="379020">in</span> <span m="379230">M</span> <span m="379610">and</span> <span m="379670">another</span> <span m="379980">time</span> <span m="380260">because</span> <span m="380540">it's in</span> <span m="380710">E--</span> <span m="381200">so</span> <span m="381380">it's</span> <span m="381570">twice--</span> <span m="383360">this</span> <span m="383550">particular</span> <span m="384200">area</span> <span m="384700">is</span> <span m="384870">only</span> <span m="385090">counted</span> <span m="385480">once</span> <span m="385820">in</span> <span m="385930">this</span> <span m="386040">sum</span> <span m="386410">over</span> <span m="386650">here.</span> <span m="388080">And</span> <span m="388180">similarly,</span> <span m="388740">we</span> <span m="388750">have</span> <span m="388990">two</span> <span m="389200">times</span> <span m="389580">over</span> <span m="389840">here,</span> <span m="390100">two</span> <span m="390380">times</span> <span m="390770">here,</span> <span m="391160">and</span> <span m="391360">one</span> <span m="391610">times</span> <span m="391950">over</span> <span m="392200">here.</span> <span m="392980">And</span> <span m="393210">this</span> <span m="393910">intersection</span> <span m="394710">of</span> <span m="395050">all</span> <span m="395390">of the</span> <span m="395730">three,</span> <span m="396840">well, it's</span> <span m="397060">counted</span> <span m="397920">when</span> <span m="398120">we</span> <span m="398590">count</span> <span m="398850">elements</span> <span m="399250">in</span> <span m="399360">M</span> <span m="399580">but</span> <span m="399720">also</span> <span m="400300">when</span> <span m="400430">we</span> <span m="400560">count</span> <span m="400830">elements</span> <span m="401180">in</span> <span m="401270">E and</span> <span m="401620">also</span> <span m="402030">when we count</span> <span m="402510">elements</span> <span m="402900">in</span> <span m="402990">S.</span> <span m="403870">So this</span> <span m="404070">is</span> <span m="404230">counted</span> <span m="404610">three</span> <span m="404890">times.</span> </p>
<p><span m="407700">So</span> <span m="409280">let's</span> <span m="409510">have</span> <span m="409670">a</span> <span m="409810">look.</span> <span m="410260">It</span> <span m="410390">seems</span> <span m="410760">like</span> <span m="411570">we</span> <span m="411690">want</span> <span m="411920">to</span> <span m="412020">do</span> <span m="412270">exactly</span> <span m="412790">the</span> <span m="412890">same</span> <span m="413170">as</span> <span m="413350">what</span> <span m="413460">we</span> <span m="413520">did</span> <span m="413720">before.</span> <span m="415390">We</span> <span m="415540">like</span> <span m="415760">to</span> <span m="417100">subtract</span> <span m="417435">an</span> <span m="417770">intersection</span> <span m="418990">of</span> <span m="419270">M</span> <span m="419810">and</span> <span m="420090">S.</span> <span m="421590">That's</span> <span m="421820">this</span> <span m="422050">part.</span> <span m="422240">It</span> <span m="422390">cancels</span> <span m="422930">the</span> <span m="423040">double</span> <span m="423320">counting</span> <span m="423720">in</span> <span m="423790">this</span> <span m="424040">area,</span> <span m="424420">right?</span> <span m="425440">We</span> <span m="425550">can</span> <span m="425690">subtract</span> <span m="426550">this</span> <span m="426870">intersection</span> <span m="427760">such</span> <span m="427970">that</span> <span m="428160">this</span> <span m="428490">part</span> <span m="428820">is</span> <span m="429670">cancelled.</span> <span m="430520">The</span> <span m="430580">double</span> <span m="430850">counting</span> <span m="431200">here</span> <span m="431480">is</span> <span m="431650">canceled.</span> <span m="432340">And</span> <span m="432510">similarly,</span> <span m="432960">we want</span> <span m="433130">to</span> <span m="433530">subtract</span> <span m="434290">the</span> <span m="434430">intersection</span> <span m="434980">between</span> <span m="436120">E</span> <span m="436420">and</span> <span m="436640">S</span> <span m="437160">which</span> <span m="437500">would</span> <span m="437750">eliminate</span> <span m="438300">this</span> <span m="438530">double</span> <span m="438810">counting.</span> </p>
<p><span m="439300">But</span> <span m="439360">now</span> <span m="439530">we</span> <span m="439650">can</span> <span m="439800">see</span> <span m="440440">that</span> <span m="440610">we start</span> <span m="441120">to</span> <span m="441590">subtract</span> <span m="442090">too</span> <span m="442240">much</span> <span m="443700">because</span> <span m="445030">this</span> <span m="445560">central</span> <span m="446060">area</span> <span m="446390">here</span> <span m="447240">will</span> <span m="447430">be--</span> <span m="448730">well,</span> <span m="449030">it's</span> <span m="449430">part</span> <span m="449650">of</span> <span m="449820">each</span> <span m="450070">of</span> <span m="450190">those</span> <span m="450410">intersections</span> <span m="451000">so</span> <span m="451940">it's</span> <span m="452140">actually</span> <span m="453360">eliminated.</span> <span m="455130">All</span> <span m="455390">of</span> <span m="455510">it</span> <span m="456820">is</span> <span m="457040">subtracted.</span> </p>
<p><span m="458310">So</span> <span m="458390">let's</span> <span m="459220">create</span> <span m="459540">the</span> <span m="459620">same</span> <span m="459960">diagram.</span> <span m="461490">But</span> <span m="461640">now</span> <span m="461830">when</span> <span m="462010">we</span> <span m="462170">count</span> <span m="462840">the</span> <span m="464850">sum</span> <span m="465370">of</span> <span m="465600">all</span> <span m="465980">those</span> <span m="466350">intersections--</span> <span m="467370">so</span> <span m="467530">we</span> <span m="467690">have</span> <span m="468450">the</span> <span m="468590">intersection</span> <span m="471140">of</span> <span m="471430">M</span> <span m="472220">intersect</span> <span m="472840">with</span> <span m="473090">E,</span> <span m="474310">the</span> <span m="474780">intersection</span> <span m="475490">of</span> <span m="475970">M</span> <span m="476490">with</span> <span m="476890">S,</span> <span m="478450">and</span> <span m="478660">we</span> <span m="478830">also</span> <span m="479150">want</span> <span m="479320">to</span> <span m="479720">figure</span> <span m="480080">out</span> <span m="480240">how</span> <span m="480360">many</span> <span m="480670">times</span> <span m="481130">elements</span> <span m="481480">are counted</span> <span m="482050">in</span> <span m="483210">the</span> <span m="483320">set</span> <span m="485810">E</span> <span m="486130">intersects</span> <span m="486770">with</span> <span m="487010">S.</span> <span m="489570">So</span> <span m="489880">let's</span> <span m="490060">have</span> <span m="490260">a look.</span> </p>
<p><span m="490490">Well,</span> <span m="491640">this</span> <span m="491990">area</span> <span m="492320">here is</span> <span m="492620">counted</span> <span m="492870">zero</span> <span m="493250">times.</span> <span m="493830">It's</span> <span m="494040">not</span> <span m="494310">part</span> <span m="494840">of</span> <span m="495290">any</span> <span m="495700">of</span> <span m="495880">these</span> <span m="496090">intersections</span> <span m="497330">because</span> <span m="497760">it--</span> <span m="498570">this</span> <span m="498740">part</span> <span m="498950">is</span> <span m="499160">only</span> <span m="499510">in</span> <span m="499670">M</span> <span m="500376">and</span> <span m="500730">not</span> <span m="501000">in</span> <span m="501180">S</span> <span m="501450">or</span> <span m="501670">E.</span> <span m="501950">So</span> <span m="502130">this</span> <span m="502320">is</span> <span m="502870">counted</span> <span m="503170">zero</span> <span m="503630">times.</span> <span m="505020">And</span> <span m="505190">here</span> <span m="505420">we</span> <span m="505530">have</span> <span m="505790">the</span> <span m="505900">same.</span> <span m="507530">Well,</span> <span m="507660">this</span> <span m="507870">part</span> <span m="508070">over</span> <span m="508330">here</span> <span m="508530">is</span> <span m="508640">exactly</span> <span m="509200">once</span> <span m="509530">in</span> <span m="509660">this</span> <span m="509860">intersection</span> <span m="510540">between</span> <span m="510840">those</span> <span m="511080">two</span> <span m="511220">sets.</span> <span m="511980">It's</span> <span m="512179">not</span> <span m="512409">in</span> <span m="512490">this</span> <span m="512650">one so this</span> <span m="513130">is counted</span> <span m="513730">once.</span> <span m="515710">So</span> <span m="516030">we</span> <span m="516510">have</span> <span m="516760">by</span> <span m="516870">symmetry</span> <span m="517460">that</span> <span m="517690">for</span> <span m="517840">all</span> <span m="517990">the</span> <span m="518130">others.</span> </p>
<p><span m="519030">This</span> <span m="519340">area,</span> <span m="519710">however,</span> <span m="520770">is</span> <span m="520990">in</span> <span m="521159">both</span> <span m="521549">this</span> <span m="521860">intersection</span> <span m="522150">as</span> <span m="522440">well as</span> <span m="522809">this</span> <span m="523030">one</span> <span m="523289">and</span> <span m="523549">also</span> <span m="523770">this</span> <span m="524030">one.</span> <span m="524200">So</span> <span m="524310">it's</span> <span m="524450">counted</span> <span m="524720">three</span> <span m="525050">times.</span> <span m="527270">So</span> <span m="527440">now</span> <span m="527630">we</span> <span m="527760">can</span> <span m="527960">easily</span> <span m="529240">start</span> <span m="529620">to</span> <span m="529710">combine</span> <span m="530450">all</span> <span m="530750">these</span> <span m="531040">together.</span> <span m="531540">And</span> <span m="531670">you</span> <span m="531760">can</span> <span m="531940">see,</span> <span m="532610">if</span> <span m="532880">we</span> <span m="534160">take</span> <span m="535240">this</span> <span m="535590">sum</span> <span m="536540">and</span> <span m="536690">we</span> <span m="536790">subtract</span> <span m="537350">this,</span> <span m="538490">then</span> <span m="538660">the</span> <span m="538730">middle</span> <span m="539050">area</span> <span m="539280">is</span> <span m="539490">cancelled.</span> <span m="540020">So</span> <span m="540100">we</span> <span m="540210">have</span> <span m="540420">to</span> <span m="540510">still</span> <span m="540840">add</span> <span m="541080">the</span> <span m="541220">intersection</span> <span m="541850">of</span> <span m="541980">the</span> <span m="542080">three</span> <span m="542390">sets.</span> <span m="543180">The</span> <span m="543270">intersection</span> <span m="543860">of</span> <span m="544040">M,</span> <span m="544892">E</span> <span m="545320">and</span> <span m="545530">S</span> <span m="545890">together.</span> </p>
<p><span m="547030">So</span> <span m="547230">in</span> <span m="547330">a</span> <span m="547380">formula</span> <span m="548080">that</span> <span m="548250">would</span> <span m="549210">look</span> <span m="549450">like</span> <span m="549630">this.</span> <span m="551060">So</span> <span m="553620">we</span> <span m="553770">are</span> <span m="553930">interested</span> <span m="554540">in</span> <span m="554630">the</span> <span m="554730">union.</span> <span m="557930">And</span> <span m="558060">this</span> <span m="558230">can</span> <span m="558370">now</span> <span m="558520">be</span> <span m="558690">written</span> <span m="559040">as,</span> <span m="560340">well,</span> <span m="560540">the</span> <span m="560630">sum</span> <span m="561350">of</span> <span m="561830">the</span> <span m="561950">cardinalities</span> <span m="562690">of the</span> <span m="562820">individual</span> <span m="563370">sets.</span> <span m="566000">And</span> <span m="566590">we</span> <span m="567240">subtract</span> <span m="571970">the</span> <span m="572310">intersections</span> <span m="572830">that</span> <span m="572980">we</span> <span m="573090">have</span> <span m="573480">up there.</span> <span m="574160">So</span> <span m="574530">M</span> <span m="574700">intersect</span> <span m="575340">E</span> <span m="576260">minus</span> <span m="577560">M</span> <span m="578080">intersect</span> <span m="578743">with</span> <span m="579146">S</span> <span m="580100">and</span> <span m="580280">minus</span> <span m="580780">E</span> <span m="581060">intersect</span> <span m="581740">with</span> <span m="582550">S.</span> </p>
<p><span m="584780">Now</span> <span m="584980">we</span> <span m="585160">have</span> <span m="586250">eliminated--</span> <span m="587250">if</span> <span m="587460">you</span> <span m="587560">look</span> <span m="587730">at</span> <span m="587780">this</span> <span m="587970">Venn</span> <span m="588140">diagram</span> <span m="588540">over here--</span> <span m="588920">this</span> <span m="589050">diagram</span> <span m="589420">over</span> <span m="589570">here-- we've</span> <span m="589940">eliminated</span> <span m="590480">everything</span> <span m="590850">that's</span> <span m="591050">in</span> <span m="591130">the</span> <span m="591210">middle</span> <span m="591960">area.</span> <span m="592950">So</span> <span m="593150">we</span> <span m="593610">still</span> <span m="593980">need</span> <span m="594320">to add</span> <span m="594720">intersection--</span> <span m="595510">everything</span> <span m="595940">that</span> <span m="596130">is</span> <span m="596490">counted</span> <span m="596900">in</span> <span m="597000">the</span> <span m="597090">intersection</span> <span m="597650">of</span> <span m="597830">M,</span> <span m="598556">E,</span> <span m="599860">as</span> <span m="600070">well</span> <span m="600470">as</span> <span m="600890">S.</span> <span m="602220">So</span> <span m="602420">we have</span> <span m="602630">this</span> <span m="602810">equation.</span> <span m="603940">And</span> <span m="604690">now</span> <span m="604840">you</span> <span m="604950">can</span> <span m="605080">see</span> <span m="605350">why</span> <span m="605750">we</span> <span m="605980">call</span> <span m="606190">this</span> <span m="606320">the</span> <span m="606400">inclusion-exclusion</span> <span m="607390">principle.</span> </p>
<p><span m="608700">Here</span> <span m="608960">we</span> <span m="609820">include</span> <span m="612080">all</span> <span m="613250">the</span> <span m="613260">elements</span> <span m="613730">in</span> <span m="614150">M, E</span> <span m="614555">and S.</span> <span m="615510">Now</span> <span m="615760">we</span> <span m="615870">have</span> <span m="616060">to</span> <span m="616230">exclude</span> <span m="616930">what we</span> <span m="617230">counted</span> <span m="617610">too</span> <span m="617790">much.</span> <span m="618800">So</span> <span m="619010">we</span> <span m="619110">have</span> <span m="619360">exclusion</span> <span m="620030">over</span> <span m="620310">here.</span> <span m="622310">And</span> <span m="622420">now</span> <span m="622750">we,</span> <span m="623180">again,</span> <span m="623520">have</span> <span m="623650">to</span> <span m="623740">include</span> <span m="624210">some</span> <span m="624400">stuff</span> <span m="624830">because</span> <span m="625880">we</span> <span m="626090">excluded</span> <span m="626600">too</span> <span m="626750">much.</span> <span m="629080">And</span> <span m="629280">then</span> <span m="629390">we</span> <span m="629490">keep</span> <span m="629750">on</span> <span m="629860">going</span> <span m="630140">like</span> <span m="630370">this.</span> </p>
<p><span m="631430">So,</span> <span m="631650">in</span> <span m="631750">general,</span> <span m="632440">we</span> <span m="632820">get</span> <span m="633130">the</span> <span m="633210">following</span> <span m="637290">theorem</span> <span m="639420">that</span> <span m="639620">says</span> <span m="639970">that,</span> <span m="640360">if</span> <span m="640580">you</span> <span m="640720">look</span> <span m="640930">at</span> <span m="641070">the</span> <span m="641170">cardinalities--</span> <span m="643640">of</span> <span m="643810">the</span> <span m="643900">cardinality</span> <span m="644430">of</span> <span m="644550">the</span> <span m="644650">union</span> <span m="645260">of</span> <span m="647200">a</span> <span m="647260">whole</span> <span m="647550">bunch</span> <span m="647820">of</span> <span m="647890">sets--</span> <span m="648770">say we</span> <span m="649130">have</span> <span m="651060">n</span> <span m="651340">sets--</span> <span m="652570">then</span> <span m="652740">this</span> <span m="652920">is</span> <span m="653050">actually</span> <span m="653470">equal</span> <span m="653820">to--</span> <span m="654380">well,</span> <span m="654490">we</span> <span m="654620">first</span> <span m="655330">include</span> <span m="656290">the</span> <span m="656410">sums</span> <span m="657070">of</span> <span m="657200">all</span> <span m="657490">the</span> <span m="657580">individual</span> <span m="658010">cardinalities.</span> <span m="659900">Which</span> <span m="660020">is</span> <span m="661530">from</span> <span m="661740">I</span> <span m="661950">equals</span> <span m="662280">1</span> <span m="662490">to</span> <span m="662700">n</span> <span m="664990">cardinality</span> <span m="665650">of</span> <span m="665800">Ai</span> <span m="667510">minus--</span> <span m="668780">now</span> <span m="668960">we</span> <span m="669050">have</span> <span m="669250">to,</span> <span m="670290">again,</span> <span m="670580">we have to</span> <span m="670750">eliminate</span> <span m="671270">all the</span> <span m="671350">double</span> <span m="671690">counting--</span> <span m="672770">so</span> <span m="672960">we</span> <span m="673120">look</span> <span m="673460">at</span> <span m="674780">every</span> <span m="675165">pair</span> <span m="675550">of</span> <span m="675620">sets</span> <span m="676820">indexed</span> <span m="677300">by</span> <span m="677530">i1</span> <span m="677730">and</span> <span m="678120">i2.</span> <span m="679580">And</span> <span m="679770">we</span> <span m="680680">look</span> <span m="680940">at</span> <span m="681200">their</span> <span m="681390">intersection</span> <span m="682690">and</span> <span m="682890">we</span> <span m="683930">take</span> <span m="684200">that</span> <span m="684410">out</span> <span m="685030">using</span> <span m="685370">this</span> <span m="685760">minus</span> <span m="686120">sign.</span> <span m="686320">We</span> <span m="686430">exclude</span> <span m="686930">this.</span> </p>
<p><span m="690240">And</span> <span m="690360">we</span> <span m="690400">continue</span> <span m="690860">like</span> <span m="691090">this.</span> <span m="691350">So</span> <span m="691570">we</span> <span m="692110">add</span> <span m="692390">now--</span> <span m="693380">because</span> <span m="693790">we have</span> <span m="693950">excluded</span> <span m="694860">too</span> <span m="695020">much,</span> <span m="695250">we have to</span> <span m="695530">include</span> <span m="695900">some</span> <span m="696090">more--</span> <span m="699730">we</span> <span m="699850">look</span> <span m="700000">at</span> <span m="700090">all</span> <span m="700310">triples</span> <span m="700630">of sets</span> <span m="702050">indexed</span> <span m="702460">by</span> <span m="702670">i1,</span> <span m="703170">i2--</span> <span m="704080">oops--</span> <span m="705860">and</span> <span m="706460">i3.</span> <span m="709820">And</span> <span m="710760">now</span> <span m="710970">we</span> <span m="711120">look</span> <span m="711380">at</span> <span m="712560">the</span> <span m="712680">intersection</span> <span m="713200">of</span> <span m="713360">those</span> <span m="713620">sets.</span> <span m="718400">And</span> <span m="718550">we</span> <span m="718590">continue</span> <span m="719440">like</span> <span m="719670">this</span> <span m="720290">all</span> <span m="720630">the</span> <span m="720730">way</span> <span m="722030">until</span> <span m="723960">we</span> <span m="724150">look</span> <span m="724450">at</span> <span m="726770">the</span> <span m="727120">very</span> <span m="727380">last</span> <span m="727740">intersection</span> <span m="728360">that</span> <span m="728510">you</span> <span m="728600">can</span> <span m="728750">have,</span> <span m="729170">which</span> <span m="729310">is</span> <span m="729770">intersection</span> <span m="730330">of</span> <span m="730430">all</span> <span m="730720">the</span> <span m="730800">sets.</span> <span m="732830">And</span> <span m="733470">it</span> <span m="733610">depends</span> <span m="734110">on</span> <span m="734290">whether</span> <span m="736230">n</span> <span m="736440">is</span> <span m="736600">even</span> <span m="736950">or</span> <span m="737110">add</span> <span m="737450">whether we</span> <span m="737790">have</span> <span m="737970">to</span> <span m="738150">include</span> <span m="738680">or</span> <span m="738800">exclude</span> <span m="739310">it.</span> </p>
<p><span m="740040">So</span> <span m="740250">we</span> <span m="740470">have a</span> <span m="740520">minus</span> <span m="741020">1</span> <span m="741290">to</span> <span m="741400">the</span> <span m="741510">power</span> <span m="741920">n</span> <span m="742100">plus</span> <span m="742410">1.</span> <span m="744520">And</span> <span m="744730">then</span> <span m="745590">we</span> <span m="745710">have</span> <span m="746010">the</span> <span m="746060">intersection</span> <span m="746910">of</span> <span m="747070">all</span> <span m="747430">of</span> <span m="747550">these.</span> <span m="749950">Now,</span> <span m="750070">we</span> <span m="750170">can</span> <span m="750320">write</span> <span m="750590">this</span> <span m="750840">out</span> <span m="751645">in</span> <span m="752060">a</span> <span m="752130">more</span> <span m="752370">elegant</span> <span m="752780">formula</span> <span m="754900">where</span> <span m="756040">we</span> <span m="756130">just</span> <span m="756440">put</span> <span m="757050">all</span> <span m="757280">this</span> <span m="757830">what</span> <span m="758040">I</span> <span m="758110">wrote</span> <span m="758380">here</span> <span m="758770">in</span> <span m="759260">a</span> <span m="759390">sum.</span> <span m="760690">And</span> <span m="760850">then</span> <span m="760970">it</span> <span m="761060">looks</span> <span m="761290">like</span> <span m="761480">this.</span> <span m="762010">It's</span> <span m="762120">the</span> <span m="762180">sum</span> <span m="762530">over</span> <span m="763630">k</span> <span m="764010">equals</span> <span m="764460">1</span> <span m="765170">to</span> <span m="765390">n</span> <span m="766640">minus</span> <span m="767060">1</span> <span m="767310">to</span> <span m="767430">the</span> <span m="767540">power</span> <span m="767770">k</span> <span m="768080">plus</span> <span m="768410">1.</span> <span m="769460">So</span> <span m="770060">depends</span> <span m="770240">whether</span> <span m="770600">k</span> <span m="770750">is even</span> <span m="771035">or</span> <span m="771320">odd</span> <span m="771540">whether</span> <span m="772020">we</span> <span m="772110">do</span> <span m="772330">an</span> <span m="772460">inclusion</span> <span m="773010">or an</span> <span m="773190">exclusion</span> <span m="773545">in</span> <span m="773900">this</span> <span m="773990">sum.</span> </p>
<p><span m="775570">And</span> <span m="776200">we</span> <span m="776370">look</span> <span m="776590">at</span> <span m="776720">all</span> <span m="776990">the</span> <span m="777070">subsets</span> <span m="778870">that</span> <span m="779040">our</span> <span m="781870">subsets</span> <span m="782560">of</span> <span m="782845">the</span> <span m="783130">index</span> <span m="783870">set</span> <span m="784350">1</span> <span m="784680">up</span> <span m="784870">to</span> <span m="785050">n</span> <span m="786860">such</span> <span m="787280">that</span> <span m="789210">the</span> <span m="789330">cardinality</span> <span m="790140">of</span> <span m="790570">the</span> <span m="790680">set</span> <span m="790960">S</span> <span m="791220">is</span> <span m="791400">equal</span> <span m="791620">to</span> <span m="791820">k.</span> <span m="793440">And</span> <span m="793690">now</span> <span m="793840">we</span> <span m="793940">need</span> <span m="794150">to</span> <span m="794880">look</span> <span m="795160">at</span> <span m="795240">the</span> <span m="795290">cardinality</span> <span m="796250">of</span> <span m="796600">all--</span> <span m="798430">of</span> <span m="798460">the</span> <span m="798680">intersection</span> <span m="799350">of</span> <span m="799590">all</span> <span m="800010">the</span> <span m="800100">sets</span> <span m="800570">that</span> <span m="800770">are</span> <span m="801390">indexed</span> <span m="802020">by</span> <span m="803300">S.</span> <span m="803880">So</span> <span m="804090">that's--</span> <span m="805870">so</span> <span m="806110">I</span> <span m="806210">run--</span> <span m="807560">I</span> <span m="807640">put</span> <span m="807930">i--</span> <span m="809630">so</span> <span m="809780">do the</span> <span m="810110">intersection</span> <span m="810520">of</span> <span m="810620">all</span> <span m="810850">i</span> <span m="811265">in S</span> <span m="811680">of</span> <span m="812030">the</span> <span m="812600">sets</span> <span m="813150">Ai.</span> </p>
<p><span m="814630">So</span> <span m="814820">this</span> <span m="815510">general</span> <span m="815830">formula</span> <span m="816190">is the</span> <span m="816550">inclusion-exclusion</span> <span m="817500">principle.</span> <span m="819430">And</span> <span m="819600">this</span> <span m="819790">is</span> <span m="819880">the</span> <span m="820090">way</span> <span m="820270">how</span> <span m="820410">we</span> <span m="820510">can</span> <span m="820660">think</span> <span m="820930">about</span> <span m="821250">it</span> <span m="821350">how</span> <span m="821490">to</span> <span m="821620">derive</span> <span m="821900">it.</span> <span m="822980">Of</span> <span m="823150">course,</span> <span m="823590">this</span> <span m="823740">is</span> <span m="823870">not</span> <span m="824030">a proof.</span> <span m="825140">But</span> <span m="825570">you</span> <span m="825750">could</span> <span m="825920">do</span> <span m="826060">this</span> <span m="826240">yourself</span> <span m="826750">by</span> <span m="826870">using</span> <span m="827060">induction.</span> <span m="828250">And</span> <span m="828600">you</span> <span m="828690">can</span> <span m="829020">use</span> <span m="829300">induction</span> <span m="829680">on</span> <span m="829820">n.</span> <span m="830700">So</span> <span m="830960">that's</span> <span m="832540">a</span> <span m="832950">pretty</span> <span m="833190">straightforward</span> <span m="833770">exercise.</span> </p>
<p><span m="835400">OK.</span> <span m="835750">So</span> <span m="836230">let's</span> <span m="836750">give</span> <span m="836920">an</span> <span m="837010">example.</span> <span m="840180">Let's</span> <span m="841020">do</span> <span m="841170">that</span> <span m="841940">over</span> <span m="842270">here.</span> <span m="844630">You</span> <span m="844810">will</span> <span m="844960">have</span> <span m="845250">a</span> <span m="845350">problem</span> <span m="845710">that's</span> <span m="845930">similar</span> <span m="846490">on</span> <span m="846650">the</span> <span m="846720">problem</span> <span m="847100">set.</span> <span m="850290">And</span> <span m="850510">it's</span> <span m="850640">goes</span> <span m="850910">like</span> <span m="851090">this.</span> <span m="856190">You</span> <span m="856340">would</span> <span m="856420">like</span> <span m="856640">to</span> <span m="856770">know</span> <span m="857300">how</span> <span m="857510">many</span> <span m="857810">permutations</span> <span m="858720">are</span> <span m="858920">there</span> <span m="860540">with</span> <span m="860780">a</span> <span m="860900">special</span> <span m="861290">property.</span> <span m="861740">So</span> <span m="861890">how</span> <span m="862060">many</span> <span m="862450">permutations</span> <span m="865080">do</span> <span m="865230">I</span> <span m="865380">have</span> <span m="869600">of</span> <span m="870490">the</span> <span m="870680">set</span> <span m="871260">0,</span> <span m="871930">1</span> <span m="873120">all</span> <span m="873290">the</span> <span m="873350">way</span> <span m="873510">up</span> <span m="873610">to</span> <span m="873730">9.</span> <span m="875420">So</span> <span m="876270">all the</span> <span m="876350">digits.</span> <span m="878120">And</span> <span m="879740">that</span> <span m="879950">have</span> <span m="883070">consecutive--</span> <span m="884160">so</span> <span m="884650">I put</span> <span m="885060">this</span> <span m="885540">in brackets</span> <span m="887000">just</span> <span m="887140">to</span> <span m="887230">clarify--</span> <span m="888700">so</span> <span m="888950">consecutive</span> <span m="889640">either</span> <span m="890630">a 4 and</span> <span m="891040">a</span> <span m="891270">2</span> <span m="891610">right</span> <span m="891880">next</span> <span m="892080">to</span> <span m="892170">one</span> <span m="892320">another.</span> <span m="892570">So</span> <span m="892680">remember,</span> <span m="893020">a</span> <span m="893080">permutation</span> <span m="893780">is</span> <span m="893870">a</span> <span m="893970">sequence</span> <span m="895900">that</span> <span m="896070">has</span> <span m="896680">each</span> <span m="896940">of</span> <span m="897050">those</span> <span m="897540">digits</span> <span m="898090">exactly</span> <span m="898620">once.</span> </p>
<p><span m="899900">And</span> <span m="901240">I</span> <span m="901430">like to</span> <span m="901750">count</span> <span m="902035">those</span> <span m="902320">permutations</span> <span m="903260">that</span> <span m="903420">have</span> <span m="903950">a</span> <span m="904070">4</span> <span m="904420">and a</span> <span m="904630">2</span> <span m="904780">next</span> <span m="905070">to</span> <span m="905160">one</span> <span m="905310">another</span> <span m="905640">in</span> <span m="905730">this</span> <span m="905950">way.</span> <span m="906980">So</span> <span m="907190">we</span> <span m="907310">have</span> <span m="907480">4,</span> <span m="907780">2.</span> <span m="908270">Or</span> <span m="909230">the</span> <span m="909340">permutation</span> <span m="910020">may</span> <span m="910200">have</span> <span m="911476">a</span> <span m="911860">0, 4</span> <span m="913450">or</span> <span m="915950">a</span> <span m="916020">6,</span> <span m="916340">0.</span> <span m="916800">So</span> <span m="916960">I'm very</span> <span m="917290">curious,</span> <span m="917730">how</span> <span m="917850">many</span> <span m="918110">permutations</span> <span m="918780">do</span> <span m="918950">I</span> <span m="919030">have?</span> <span m="919725">And I'm going</span> <span m="920010">to</span> <span m="920140">use</span> <span m="920310">this</span> <span m="920470">principle</span> <span m="921300">that</span> <span m="921430">I</span> <span m="921560">just</span> <span m="921810">described.</span> </p>
<p><span m="924730">So</span> <span m="926550">how</span> <span m="926690">do</span> <span m="926790">we</span> <span m="926880">need</span> <span m="927050">to</span> <span m="927140">go</span> <span m="927400">ahead</span> <span m="927700">with</span> <span m="927870">this?</span> <span m="929970">Well,</span> <span m="930180">this--</span> <span m="931170">so</span> <span m="931300">I'm</span> <span m="931410">interested</span> <span m="932010">in</span> <span m="932470">the</span> <span m="932600">permutations</span> <span m="933320">that</span> <span m="933480">have</span> <span m="933960">a 4</span> <span m="934370">and</span> <span m="934530">2</span> <span m="934710">next</span> <span m="934940">to</span> <span m="935020">one</span> <span m="935180">another.</span> <span m="936110">So</span> <span m="936250">that</span> <span m="936470">could</span> <span m="936640">be</span> <span m="936810">my</span> <span m="936960">first</span> <span m="937340">set,</span> <span m="937700">say,</span> <span m="938060">M.</span> <span m="938770">And</span> <span m="938890">then</span> <span m="939050">I</span> <span m="939140">have</span> <span m="939490">a</span> <span m="939580">set</span> <span m="940170">that</span> <span m="940390">I</span> <span m="940430">want</span> <span m="940640">to</span> <span m="940770">count</span> <span m="941190">that</span> <span m="941470">has the</span> <span m="941570">0</span> <span m="941760">and</span> <span m="941830">the</span> <span m="941910">4</span> <span m="942370">next to</span> <span m="942640">one another.</span> <span m="942860">That</span> <span m="943030">could</span> <span m="943170">be</span> <span m="943300">my</span> <span m="943510">E.</span> <span m="944110">And</span> <span m="944350">the S</span> <span m="944780">could be</span> <span m="945520">those</span> <span m="945840">permutations</span> <span m="946550">that</span> <span m="946650">have</span> <span m="946820">the</span> <span m="946940">6 and a</span> <span m="947340">0</span> <span m="947720">next</span> <span m="947970">to</span> <span m="948060">one</span> <span m="948210">another.</span> <span m="948830">So</span> <span m="948970">then</span> <span m="949150">I</span> <span m="949200">would</span> <span m="949350">like</span> <span m="949540">to</span> <span m="949660">use</span> <span m="949940">that</span> <span m="950090">rule</span> <span m="950360">over</span> <span m="950630">here.</span> </p>
<p><span m="951480">So</span> <span m="952150">I</span> <span m="952270">would</span> <span m="952470">also</span> <span m="952690">need</span> <span m="952870">to</span> <span m="953020">count</span> <span m="953350">all</span> <span m="953510">the</span> <span m="953590">intersections</span> <span m="955010">and</span> <span m="955220">so</span> <span m="955410">on.</span> <span m="955630">So</span> <span m="955830">let's</span> <span m="956280">do</span> <span m="956540">this.</span> <span m="959960">Let</span> <span m="960040">me</span> <span m="960170">first</span> <span m="960420">give</span> <span m="960560">an</span> <span m="960660">example</span> <span m="962220">of</span> <span m="962290">such</span> <span m="962550">a</span> <span m="962610">permutation.</span> <span m="963780">So</span> <span m="964600">for</span> <span m="964820">example,</span> <span m="965490">I</span> <span m="965590">could</span> <span m="965740">have the</span> <span m="966030">permutation</span> <span m="967250">that</span> <span m="967370">is the</span> <span m="967650">sequence</span> <span m="968350">7,</span> <span m="969340">2,</span> <span m="970100">5,</span> <span m="971650">6,</span> <span m="972150">0,</span> <span m="972980">and</span> <span m="973180">the 4,</span> <span m="973570">for</span> <span m="973800">example,</span> <span m="975290">3,</span> <span m="975980">5,</span> <span m="977870">1</span> <span m="978290">and</span> <span m="978460">9.</span> <span m="979420">So</span> <span m="979670">I</span> <span m="979780">have</span> <span m="980000">all</span> <span m="980200">the</span> <span m="980300">digits</span> <span m="981080">that</span> <span m="981330">are</span> <span m="981580">in</span> <span m="981730">the--</span> <span m="982280">contained</span> <span m="982720">in</span> <span m="982800">the</span> <span m="982880">set.</span> <span m="983680">And</span> <span m="984010">I</span> <span m="984060">see</span> <span m="984390">that,</span> <span m="984920">actually,</span> <span m="985490">this</span> <span m="985650">particular</span> <span m="986130">permutation</span> <span m="986940">has</span> <span m="987120">a</span> <span m="987240">6 and a</span> <span m="987620">0</span> <span m="987960">next</span> <span m="988210">to</span> <span m="988280">one</span> <span m="988440">another.</span> <span m="988760">So</span> <span m="988910">it's</span> <span m="989050">in</span> <span m="989160">this</span> <span m="989370">particular--</span> <span m="990870">the</span> <span m="990960">set</span> <span m="991120">of</span> <span m="991190">permutations</span> <span m="991900">that</span> <span m="992010">have</span> <span m="992170">a</span> <span m="992250">6 and a</span> <span m="992640">0.</span> <span m="993460">I</span> <span m="993510">can</span> <span m="993730">also</span> <span m="993920">see</span> <span m="994190">it</span> <span m="994440">has a</span> <span m="994530">0</span> <span m="994740">and</span> <span m="994880">a</span> <span m="994930">4.</span> <span m="995710">So</span> <span m="995880">it's</span> <span m="996000">also</span> <span m="996360">in</span> <span m="996440">the</span> <span m="996520">set</span> <span m="996720">of</span> <span m="996790">permutations</span> <span m="997460">that</span> <span m="997530">have</span> <span m="997830">a 0</span> <span m="998000">and</span> <span m="998090">a</span> <span m="998140">4</span> <span m="998390">next</span> <span m="998600">to</span> <span m="998680">one</span> <span m="998830">another.</span> </p>
<p><span m="1000800">OK.</span> <span m="1001060">So</span> <span m="1001190">now</span> <span m="1001340">I'm</span> <span m="1001470">going</span> <span m="1001710">to</span> <span m="1001840">define</span> <span m="1002740">these</span> <span m="1002980">different</span> <span m="1003360">sets</span> <span m="1006260">and</span> <span m="1006480">then we're</span> <span m="1006660">going</span> <span m="1006870">to</span> <span m="1006970">start</span> <span m="1007310">counting</span> <span m="1007770">these.</span> <span m="1010530">And</span> <span m="1010640">this</span> <span m="1010790">is</span> <span m="1010920">the</span> <span m="1011000">type</span> <span m="1011290">of</span> <span m="1012510">counting</span> <span m="1013150">that</span> <span m="1013310">you're</span> <span m="1013450">doing</span> <span m="1013770">in this</span> <span m="1013960">problem</span> <span m="1014250">set.</span> <span m="1015740">And</span> <span m="1016270">the</span> <span m="1016420">intuition</span> <span m="1017020">that you</span> <span m="1018330">create</span> <span m="1019210">by</span> <span m="1019320">doing</span> <span m="1019660">that</span> <span m="1020370">you</span> <span m="1020530">can</span> <span m="1020710">use</span> <span m="1021310">later</span> <span m="1021640">on</span> <span m="1022230">when</span> <span m="1022390">we</span> <span m="1022500">study</span> <span m="1022830">probability</span> <span m="1023360">theory.</span> </p>
<p><span m="1027460">OK.</span> <span m="1032460">So</span> <span m="1033010">let's</span> <span m="1033230">define</span> <span m="1034190">the</span> <span m="1034270">set</span> <span m="1034650">P4, 2</span> <span m="1036710">as</span> <span m="1037440">all</span> <span m="1037690">the</span> <span m="1037780">permutations</span> <span m="1039760">that</span> <span m="1039960">have--</span> <span m="1041380">so</span> <span m="1041710">it's the</span> <span m="1041750">set</span> <span m="1043030">of</span> <span m="1043430">permutations</span> <span m="1044430">with</span> <span m="1046619">4,</span> <span m="1047020">2.</span> <span m="1047980">And</span> <span m="1048160">similarly,</span> <span m="1048830">we</span> <span m="1048880">have</span> <span m="1049250">P0,</span> <span m="1050060">4</span> <span m="1051390">which</span> <span m="1051650">is</span> <span m="1051770">the</span> <span m="1051840">same</span> <span m="1052140">thing</span> <span m="1052475">but</span> <span m="1052810">now</span> <span m="1053380">with</span> <span m="1053560">0,</span> <span m="1053920">4.</span> <span m="1055320">P6,</span> <span m="1055650">0,</span> <span m="1057026">the</span> <span m="1057410">set</span> <span m="1057610">of</span> <span m="1057700">full</span> <span m="1057760">permutations</span> <span m="1058600">with</span> <span m="1058740">6,</span> <span m="1059040">0.</span> </p>
<p><span m="1060680">So</span> <span m="1060870">let's</span> <span m="1061070">have</span> <span m="1061250">a</span> <span m="1061330">look</span> <span m="1061780">at</span> <span m="1062080">what</span> <span m="1062270">the</span> <span m="1062360">sizes</span> <span m="1062860">of</span> <span m="1062990">an</span> <span m="1063070">individual</span> <span m="1064030">set</span> <span m="1064220">of</span> <span m="1064300">permutations.</span> <span m="1065900">How</span> <span m="1065970">can</span> <span m="1066090">we</span> <span m="1066190">do</span> <span m="1066340">this?</span> <span m="1067310">So</span> <span m="1067390">let's</span> <span m="1067580">have</span> <span m="1067750">a</span> <span m="1067820">look</span> <span m="1068060">at</span> <span m="1070690">the</span> <span m="1070910">size</span> <span m="1072280">of,</span> <span m="1072850">say,</span> <span m="1075010">this</span> <span m="1075220">one,</span> <span m="1076040">P60.</span> <span m="1078270">Well,</span> <span m="1078940">what</span> <span m="1079090">kind</span> <span m="1079270">of</span> <span m="1079390">trick</span> <span m="1079750">can I</span> <span m="1080000">do</span> <span m="1080230">in</span> <span m="1080400">order</span> <span m="1080550">to</span> <span m="1080720">count this?</span> </p>
<p><span m="1082410">So</span> <span m="1084740">we</span> <span m="1084890">have</span> <span m="1085140">learned</span> <span m="1085430">a</span> <span m="1085480">few</span> <span m="1085780">rules</span> <span m="1086280">last</span> <span m="1086790">time.</span> <span m="1087520">And</span> <span m="1088130">what</span> <span m="1088270">I</span> <span m="1088320">would</span> <span m="1088480">like</span> <span m="1088670">to</span> <span m="1088790">do</span> <span m="1089050">is,</span> <span m="1089160">well,</span> <span m="1089310">this</span> <span m="1089565">is</span> <span m="1089820">a</span> <span m="1090060">slightly</span> <span m="1090410">more</span> <span m="1090620">complex</span> <span m="1091080">structure,</span> <span m="1092460">so</span> <span m="1092700">I would</span> <span m="1092820">like</span> <span m="1093020">to</span> <span m="1093140">map</span> <span m="1093440">this</span> <span m="1093660">to</span> <span m="1093820">a</span> <span m="1093880">set</span> <span m="1095330">by</span> <span m="1095750">means</span> <span m="1096100">of a</span> <span m="1096220">bijection</span> <span m="1097220">That</span> <span m="1097320">will</span> <span m="1097420">be</span> <span m="1097540">great.</span> <span m="1098710">Then</span> <span m="1100090">to</span> <span m="1100540">do</span> <span m="1100670">some</span> <span m="1100850">kind</span> <span m="1101070">of</span> <span m="1101210">other</span> <span m="1101510">sets</span> <span m="1102480">that</span> <span m="1103010">is</span> <span m="1103120">easy to</span> <span m="1103610">count</span> <span m="1103960">for</span> <span m="1104060">me.</span> <span m="1104520">So</span> <span m="1104670">then,</span> <span m="1105210">by</span> <span m="1105310">the bijection</span> <span m="1105970">rule,</span> <span m="1106390">I will</span> <span m="1106570">be</span> <span m="1106760">able</span> <span m="1106930">to</span> <span m="1107060">figure</span> <span m="1107370">out</span> <span m="1107550">what</span> <span m="1107690">the</span> <span m="1107770">cardinalities</span> <span m="1108460">of</span> <span m="1108570">this</span> <span m="1108680">set.</span> </p>
<p><span m="1110100">So</span> <span m="1110870">the</span> <span m="1111000">trick</span> <span m="1111280">here</span> <span m="1111600">is</span> <span m="1111930">to</span> <span m="1112230">find</span> <span m="1112570">such</span> <span m="1112880">a</span> <span m="1113020">bijection.</span> <span m="1114060">And</span> <span m="1114860">the</span> <span m="1114980">idea</span> <span m="1115400">is</span> <span m="1115770">that</span> <span m="1116040">you</span> <span m="1116160">could</span> <span m="1116440">actually</span> <span m="1116850">treat</span> <span m="1117280">the</span> <span m="1117380">6</span> <span m="1117500">and the</span> <span m="1117880">0</span> <span m="1118380">as</span> <span m="1118660">one</span> <span m="1118940">unique</span> <span m="1119300">symbol.</span> <span m="1120540">So</span> <span m="1120730">let's</span> <span m="1121240">see</span> <span m="1121750">how</span> <span m="1121890">it</span> <span m="1122020">works.</span> <span m="1122350">So</span> <span m="1122900">what</span> <span m="1123050">I</span> <span m="1123090">could</span> <span m="1123290">do</span> <span m="1123510">is</span> <span m="1123620">I can</span> <span m="1124050">have,</span> <span m="1124730">from</span> <span m="1125220">the</span> <span m="1125300">set</span> <span m="1125710">of--</span> <span m="1127170">from</span> <span m="1127510">P60--</span> <span m="1128350">so</span> <span m="1128710">all</span> <span m="1128890">the</span> <span m="1129040">permutations</span> <span m="1129750">that</span> <span m="1129860">have</span> <span m="1130040">6 and a</span> <span m="1130420">0</span> <span m="1130710">next</span> <span m="1131000">to one</span> <span m="1131190">another--</span> <span m="1132300">I</span> <span m="1132450">can</span> <span m="1132650">find</span> <span m="1133130">a</span> <span m="1133320">bijection</span> <span m="1135880">to</span> <span m="1136350">the</span> <span m="1136460">permutations</span> <span m="1138840">of</span> <span m="1139110">the</span> <span m="1139210">set</span> <span m="1142670">that</span> <span m="1143390">have</span> <span m="1144170">6,</span> <span m="1144700">0</span> <span m="1145190">as</span> <span m="1145370">a</span> <span m="1145420">single</span> <span m="1145830">symbol.</span> <span m="1147400">And</span> <span m="1147620">then</span> <span m="1147800">I</span> <span m="1147920">have</span> <span m="1148070">all</span> <span m="1148250">the</span> <span m="1148360">other</span> <span m="1148530">symbols,</span> <span m="1149040">all</span> <span m="1149140">the</span> <span m="1149270">other</span> <span m="1149450">digits.</span> <span m="1149870">So</span> <span m="1150060">1,</span> <span m="1150560">2,</span> <span m="1151060">3,</span> <span m="1152060">4,</span> <span m="1153560">5,</span> <span m="1154560">7,</span> <span m="1155966">8,</span> <span m="1157540">and</span> <span m="1157720">0.</span> </p>
<p><span m="1159400">So</span> <span m="1160170">let's</span> <span m="1160570">have</span> <span m="1160770">an</span> <span m="1160840">example</span> <span m="1162300">of--</span> <span m="1163110">we</span> <span m="1163310">have</span> <span m="1163580">one</span> <span m="1163860">up</span> <span m="1164040">here,</span> <span m="1164340">actually.</span> <span m="1169890">So</span> <span m="1170090">how</span> <span m="1170590">would</span> <span m="1170790">I</span> <span m="1170840">map</span> <span m="1171090">this?</span> <span m="1172240">This would</span> <span m="1172340">be</span> <span m="1172480">mapped</span> <span m="1172800">according</span> <span m="1173180">to</span> <span m="1173310">that</span> <span m="1174680">definition.</span> <span m="1176250">To</span> <span m="1178160">7,</span> <span m="1178900">2,</span> <span m="1179480">5.</span> <span m="1180370">And</span> <span m="1180510">now</span> <span m="1180680">I</span> <span m="1180780">combine</span> <span m="1181320">6 and</span> <span m="1181690">0</span> <span m="1182040">into</span> <span m="1182380">one</span> <span m="1182670">symbol</span> <span m="1183790">in</span> <span m="1183940">my</span> <span m="1184070">sequence.</span> <span m="1185380">And</span> <span m="1185540">then</span> <span m="1185700">I</span> <span m="1185810">have</span> <span m="1186030">the</span> <span m="1186150">rest.</span> <span m="1186580">4,</span> <span m="1187420">3,</span> <span m="1188590">5,</span> <span m="1189260">1,</span> <span m="1190370">and</span> <span m="1190560">9.</span> </p>
<p><span m="1195840">Now</span> <span m="1195950">this</span> <span m="1196170">is</span> <span m="1196290">easy</span> <span m="1196670">for</span> <span m="1196800">me</span> <span m="1196980">to</span> <span m="1197130">count,</span> <span m="1197445">right?</span> <span m="1197760">So</span> <span m="1197900">this</span> <span m="1198110">is</span> <span m="1198220">a</span> <span m="1198290">bijection.</span> <span m="1200690">And</span> <span m="1201140">by</span> <span m="1201360">the</span> <span m="1201730">bijection</span> <span m="1202060">rule,</span> <span m="1202960">the</span> <span m="1203070">cardinality</span> <span m="1203710">of</span> <span m="1203810">this</span> <span m="1204010">set</span> <span m="1204180">is</span> <span m="1204320">equal</span> <span m="1204580">to</span> <span m="1205070">the</span> <span m="1205190">total</span> <span m="1205420">number</span> <span m="1205680">of</span> <span m="1205780">permutations</span> <span m="1207220">on</span> <span m="1207680">this</span> <span m="1207900">set.</span> <span m="1209270">Now, this</span> <span m="1209490">set</span> <span m="1209760">has</span> <span m="1209900">exactly</span> <span m="1210310">eight</span> <span m="1210720">elements</span> <span m="1211690">so</span> <span m="1211890">I</span> <span m="1211970">know</span> <span m="1212180">how</span> <span m="1212310">many</span> <span m="1212550">there</span> <span m="1212770">are.</span> <span m="1213820">So</span> <span m="1214810">I</span> <span m="1214930">know</span> <span m="1215130">that the</span> <span m="1215380">cardinality</span> <span m="1216200">is</span> <span m="1216490">equal</span> <span m="1217230">to</span> <span m="1217510">8</span> <span m="1217760">factorial.</span> <span m="1218740">Because there</span> <span m="1219190">are</span> <span m="1219600">8 factorial</span> <span m="1219890">permutations</span> <span m="1220620">on</span> <span m="1220740">sets</span> <span m="1221440">of</span> <span m="1221760">size</span> <span m="1222080">8.</span> <span m="1223370">We saw</span> <span m="1223620">that</span> <span m="1224010">last</span> <span m="1224320">time.</span> <span m="1225560">So</span> <span m="1225720">that's</span> <span m="1226010">the</span> <span m="1226420">trick</span> <span m="1227860">or--</span> </p>
<p><span m="1228430">AUDIENCE: Nine.</span> </p>
<p><span m="1229070">PROFESSOR: Oh,</span> <span m="1229140">is</span> <span m="1229380">it</span> <span m="1229490">nine?</span> <span m="1230025">Oh,</span> <span m="1230280">yeah, you're</span> <span m="1230550">right.</span> <span m="1233210">Great.</span> <span m="1234440">So</span> <span m="1234610">this</span> <span m="1234810">is</span> <span m="1234930">9.</span> <span m="1239460">And</span> <span m="1239650">similarly,</span> <span m="1240670">in</span> <span m="1240800">the</span> <span m="1240880">same</span> <span m="1241150">way,</span> <span m="1241710">we</span> <span m="1241720">can</span> <span m="1241870">also</span> <span m="1242070">count</span> <span m="1242320">these,</span> <span m="1242510">right?</span> <span m="1242940">We</span> <span m="1243070">can</span> <span m="1243250">treat</span> <span m="1243550">0,</span> <span m="1243880">4</span> <span m="1244210">as</span> <span m="1244320">one</span> <span m="1244500">symbol</span> <span m="1245040">or</span> <span m="1245720">4,</span> <span m="1246130">2 as</span> <span m="1246540">one</span> <span m="1246750">symbol.</span> <span m="1247700">So</span> <span m="1247880">I</span> <span m="1248070">also</span> <span m="1248320">know</span> <span m="1248540">that</span> <span m="1248680">the</span> <span m="1248770">cardinalities</span> <span m="1249500">of</span> <span m="1249650">these</span> <span m="1250390">are</span> <span m="1250600">also</span> <span m="1251730">equal</span> <span m="1252130">to</span> <span m="1252300">9</span> <span m="1253000">factorial.</span> </p>
<p><span m="1255490">Now,</span> <span m="1255550">when</span> <span m="1255710">I</span> <span m="1255790">use the</span> <span m="1256020">inclusion-exclusion</span> <span m="1256970">principle--</span> <span m="1257560">so</span> <span m="1257790">now I've</span> <span m="1258060">computed</span> <span m="1258530">those</span> <span m="1258880">three,</span> <span m="1259190">essentially--</span> <span m="1260230">I</span> <span m="1260330">need</span> <span m="1260480">to</span> <span m="1260570">subtract</span> <span m="1261070">their</span> <span m="1261190">intersections.</span> <span m="1262240">So</span> <span m="1262940">let's</span> <span m="1263140">compute</span> <span m="1263995">the</span> <span m="1264280">intersections.</span> <span m="1270500">And</span> <span m="1272910">if</span> <span m="1273090">you</span> <span m="1273180">do</span> <span m="1273390">that,</span> <span m="1275400">we're</span> <span m="1275780">going to use</span> <span m="1275960">the</span> <span m="1276310">exact same</span> <span m="1276700">trick.</span> <span m="1277090">So</span> <span m="1277290">we're</span> <span m="1277400">going</span> <span m="1277630">to</span> <span m="1279560">find</span> <span m="1279890">the</span> <span m="1279970">bijection</span> <span m="1280520">to</span> <span m="1280660">permutations</span> <span m="1281580">of</span> <span m="1282000">a</span> <span m="1282110">set</span> <span m="1282400">of</span> <span m="1282490">symbols.</span> <span m="1283640">And we</span> <span m="1283810">just</span> <span m="1284060">have</span> <span m="1284170">to</span> <span m="1284280">specify</span> <span m="1284930">those.</span> </p>
<p><span m="1287280">So</span> <span m="1287570">let's</span> <span m="1288160">do</span> <span m="1288310">this</span> <span m="1288510">together</span> <span m="1289680">and</span> <span m="1290060">take,</span> <span m="1290730">for</span> <span m="1290860">example,</span> <span m="1291470">the</span> <span m="1291570">intersection</span> <span m="1292270">of</span> <span m="1296170">P42</span> <span m="1299060">with</span> <span m="1299510">P60.</span> <span m="1301130">So</span> <span m="1301470">what kind of</span> <span m="1301840">a bijection</span> <span m="1302460">could</span> <span m="1302720">I</span> <span m="1302800">have?</span> <span m="1304450">So</span> <span m="1306620">it will</span> <span m="1306940">be</span> <span m="1307280">all--</span> <span m="1307655">to</span> <span m="1308030">all the</span> <span m="1308220">permutations</span> <span m="1310770">on</span> <span m="1312640">what</span> <span m="1312840">kind</span> <span m="1313050">of</span> <span m="1313130">a</span> <span m="1313220">set?</span> <span m="1313860">So</span> <span m="1315420">I</span> <span m="1315510">use</span> <span m="1315750">the</span> <span m="1315840">exact</span> <span m="1316190">same</span> <span m="1316450">trick</span> <span m="1316730">here.</span> <span m="1317680">So</span> <span m="1317860">I'm</span> <span m="1317940">going</span> <span m="1318270">to</span> <span m="1318860">treat</span> <span m="1319210">4 and</span> <span m="1319600">2</span> <span m="1319880">as one</span> <span m="1320380">symbol</span> <span m="1321280">and</span> <span m="1321470">6,</span> <span m="1321765">0</span> <span m="1322060">as</span> <span m="1322200">one</span> <span m="1322400">symbol.</span> </p>
<p><span m="1323700">So</span> <span m="1324680">I</span> <span m="1324800">have</span> <span m="1324960">4,</span> <span m="1325240">2,</span> <span m="1325590">6,</span> <span m="1325960">0</span> <span m="1326840">and</span> <span m="1327090">then all</span> <span m="1327190">the</span> <span m="1327320">other</span> <span m="1328140">digits.</span> <span m="1328630">So</span> <span m="1328840">it's</span> <span m="1329000">the</span> <span m="1329140">1,</span> <span m="1329600">3,</span> <span m="1330560">5,</span> <span m="1331350">7,</span> <span m="1332660">8,</span> <span m="1333180">and</span> <span m="1333230">9.</span> <span m="1334410">You</span> <span m="1334560">can</span> <span m="1334770">count</span> <span m="1335220">that</span> <span m="1335490">this</span> <span m="1335760">number</span> <span m="1336240">is</span> <span m="1336510">8.</span> <span m="1337650">So</span> <span m="1337870">by</span> <span m="1338010">the</span> <span m="1338110">bijection</span> <span m="1338640">rule,</span> <span m="1339250">we</span> <span m="1339440">have</span> <span m="1340990">that</span> <span m="1341290">the</span> <span m="1341390">cardinality</span> <span m="1341950">of</span> <span m="1342030">this</span> <span m="1342180">intersection</span> <span m="1342880">is</span> <span m="1343060">equal</span> <span m="1343280">to</span> <span m="1343480">8</span> <span m="1343730">factorial.</span> </p>
<p><span m="1345810">And</span> <span m="1345960">we</span> <span m="1346000">continue</span> <span m="1346470">like</span> <span m="1346660">this</span> <span m="1348370">to</span> <span m="1348490">compute</span> <span m="1348830">all</span> <span m="1348980">the</span> <span m="1349080">other</span> <span m="1349300">intersections.</span> <span m="1350480">And</span> <span m="1351040">they</span> <span m="1351160">do</span> <span m="1351320">look</span> <span m="1351560">a</span> <span m="1351580">little</span> <span m="1351780">bit</span> <span m="1351900">different</span> <span m="1352250">sometimes.</span> <span m="1352780">So</span> <span m="1352890">for</span> <span m="1353060">example,</span> <span m="1354900">if</span> <span m="1355120">I</span> <span m="1355200">have</span> <span m="1355650">6,</span> <span m="1355960">0</span> <span m="1357000">and</span> <span m="1357410">P</span> <span m="1357880">0,</span> <span m="1358270">4--</span> <span m="1360580">so</span> <span m="1360900">let</span> <span m="1361070">me</span> <span m="1362330">define</span> <span m="1362840">the</span> <span m="1362900">permutation</span> <span m="1363640">first--</span> <span m="1364940">or</span> <span m="1365100">the</span> <span m="1365330">bijection</span> <span m="1365710">first.</span> <span m="1368110">So</span> <span m="1368480">how</span> <span m="1368650">do</span> <span m="1368750">we</span> <span m="1368860">do</span> <span m="1369000">this</span> <span m="1369230">one?</span> <span m="1371060">Any</span> <span m="1371250">suggestions?</span> </p>
<p><span m="1374870">So</span> <span m="1375160">what</span> <span m="1375330">kind</span> <span m="1375510">of</span> <span m="1375630">a</span> <span m="1375700">set</span> <span m="1376140">can</span> <span m="1376320">I</span> <span m="1376370">find</span> <span m="1376620">a</span> <span m="1376750">permutation</span> <span m="1377350">to?</span> <span m="1378860">If</span> <span m="1379050">I</span> <span m="1379090">look--</span> <span m="1380830">so</span> <span m="1381030">over</span> <span m="1381270">here</span> <span m="1381510">I</span> <span m="1381670">have a</span> <span m="1381820">permutation</span> <span m="1382580">in</span> <span m="1382750">which</span> <span m="1383330">6</span> <span m="1383680">and</span> <span m="1383800">0</span> <span m="1384130">are</span> <span m="1384490">next</span> <span m="1384900">to</span> <span m="1384990">one</span> <span m="1385150">another.</span> <span m="1386520">But</span> <span m="1386620">this</span> <span m="1386800">is</span> <span m="1386970">also</span> <span m="1387340">a</span> <span m="1387430">permutation</span> <span m="1388240">in</span> <span m="1388430">which</span> <span m="1388820">0</span> <span m="1389050">and</span> <span m="1389240">4</span> <span m="1389550">are</span> <span m="1389630">next</span> <span m="1389950">to one</span> <span m="1390370">another.</span> </p>
<p><span m="1390790">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="1392890">PROFESSOR: Yeah,</span> <span m="1393210">exactly.</span> <span m="1393740">So</span> <span m="1394020">3,</span> <span m="1394220">6,</span> <span m="1394950">0</span> <span m="1395340">and 4 is</span> <span m="1395720">one</span> <span m="1395890">symbol</span> <span m="1396300">because</span> <span m="1398200">permutation</span> <span m="1398900">is</span> <span m="1399070">right</span> <span m="1399390">in</span> <span m="1399540">here</span> <span m="1400140">has</span> <span m="1400750">a</span> <span m="1400910">6</span> <span m="1401290">and</span> <span m="1401490">0</span> <span m="1401800">next</span> <span m="1402080">to one</span> <span m="1402360">another.</span> <span m="1402740">And</span> <span m="1402890">the</span> <span m="1403070">0</span> <span m="1403500">should</span> <span m="1403710">also</span> <span m="1403920">be</span> <span m="1404050">next</span> <span m="1404280">to</span> <span m="1404350">the</span> <span m="1404440">4</span> <span m="1404960">because</span> <span m="1405270">it's</span> <span m="1405420">in</span> <span m="1405530">this</span> <span m="1405650">set</span> <span m="1405950">as</span> <span m="1406080">well.</span> <span m="1407180">So</span> <span m="1407390">we</span> <span m="1407510">know</span> <span m="1407780">that</span> <span m="1407990">every</span> <span m="1408210">permutation</span> <span m="1408900">in</span> <span m="1409040">here</span> <span m="1409930">has</span> <span m="1410630">a</span> <span m="1410770">sub</span> <span m="1411010">sequence</span> <span m="1411540">of</span> <span m="1411630">6, 0,</span> <span m="1412060">4</span> <span m="1413460">next</span> <span m="1413660">to</span> <span m="1413740">one</span> <span m="1413870">another.</span> <span m="1414250">So</span> <span m="1414270">we</span> <span m="1414360">can</span> <span m="1414490">treat</span> <span m="1414700">this</span> <span m="1414860">as</span> <span m="1415000">one</span> <span m="1415230">symbol.</span> <span m="1416710">And</span> <span m="1416840">then</span> <span m="1416970">we</span> <span m="1417050">have</span> <span m="1417260">all</span> <span m="1417410">the</span> <span m="1417520">other</span> <span m="1417710">ones.</span> </p>
<p><span m="1423010">And</span> <span m="1423210">again,</span> <span m="1423900">this</span> <span m="1424210">is--</span> <span m="1425100">these</span> <span m="1425445">are</span> <span m="1425790">8</span> <span m="1426030">elements.</span> <span m="1427520">So</span> <span m="1428270">we</span> <span m="1428420">have</span> <span m="1428700">that</span> <span m="1428930">also,</span> <span m="1429740">for</span> <span m="1429890">this</span> <span m="1430080">intersection,</span> <span m="1431330">the</span> <span m="1431430">cardinality</span> <span m="1432200">is</span> <span m="1432460">equal,</span> <span m="1432920">by</span> <span m="1433080">the bijection</span> <span m="1433580">rule,</span> <span m="1434340">as</span> <span m="1434780">the</span> <span m="1434920">total</span> <span m="1435200">number</span> <span m="1435240">of</span> <span m="1435310">permutations</span> <span m="1436460">on</span> <span m="1437380">a</span> <span m="1437480">set</span> <span m="1437680">of</span> <span m="1437800">eight</span> <span m="1438030">elements,</span> <span m="1438450">which</span> <span m="1438620">is</span> <span m="1438810">8</span> <span m="1439040">factorial.</span> </p>
<p><span m="1448900">So</span> <span m="1450470">for</span> <span m="1450690">the</span> <span m="1450810">last</span> <span m="1451150">one,</span> <span m="1451690">we</span> <span m="1451890">also</span> <span m="1452320">have</span> <span m="1452680">similar--</span> <span m="1455660">so</span> <span m="1455930">let</span> <span m="1456320">me</span> <span m="1456430">see</span> <span m="1456640">which</span> <span m="1456830">one</span> <span m="1457000">I</span> <span m="1457050">need</span> <span m="1457240">to</span> <span m="1457360">do.</span> <span m="1457680">So</span> <span m="1457960">I</span> <span m="1458060">have</span> <span m="1458830">4,</span> <span m="1459340">2</span> <span m="1460860">and</span> <span m="1461900">0,</span> <span m="1462310">4.</span> <span m="1463460">Well,</span> <span m="1463640">again,</span> <span m="1464830">I</span> <span m="1464960">do</span> <span m="1465090">the</span> <span m="1465200">same</span> <span m="1465500">trick.</span> <span m="1466900">Now</span> <span m="1467180">I</span> <span m="1467270">have</span> <span m="1467460">a</span> <span m="1467540">symbol</span> <span m="1467940">that</span> <span m="1468160">has--</span> <span m="1468445">well,</span> <span m="1468730">the</span> <span m="1468830">permutation</span> <span m="1469600">here</span> <span m="1469820">has</span> <span m="1470180">a</span> <span m="1470250">0</span> <span m="1470760">and</span> <span m="1470850">a</span> <span m="1470920">4</span> <span m="1471160">next</span> <span m="1471380">to</span> <span m="1471460">one</span> <span m="1471690">another,</span> <span m="1472280">and</span> <span m="1472510">the</span> <span m="1472580">4 is</span> <span m="1472930">next</span> <span m="1474010">to</span> <span m="1474500">a</span> <span m="1474720">2.</span> <span m="1475530">So</span> <span m="1475740">I</span> <span m="1475820">treat</span> <span m="1476150">this</span> <span m="1477070">as</span> <span m="1477280">one</span> <span m="1477500">symbol.</span> </p>
<p><span m="1478780">So</span> <span m="1478970">if</span> <span m="1479070">the</span> <span m="1479250">permutations</span> <span m="1480460">on</span> <span m="1480710">the</span> <span m="1480800">set</span> <span m="1483020">0,</span> <span m="1483750">4,</span> <span m="1484160">2,</span> <span m="1484840">and</span> <span m="1485020">then</span> <span m="1485260">1,</span> <span m="1485690">3,</span> <span m="1486410">5,</span> <span m="1487200">7--</span> <span m="1488730">oh,</span> <span m="1489000">and 6 as</span> <span m="1489480">well--</span> <span m="1490290">8</span> <span m="1491050">and</span> <span m="1491240">nine.</span> <span m="1492450">So</span> <span m="1492680">the</span> <span m="1492770">cardinality</span> <span m="1493530">of</span> <span m="1493650">this</span> <span m="1493860">set</span> <span m="1494930">is</span> <span m="1495300">also</span> <span m="1495690">equal</span> <span m="1496830">to</span> <span m="1498690">8</span> <span m="1498940">factorial.</span> <span m="1502140">So</span> <span m="1502230">now</span> <span m="1503320">we</span> <span m="1503420">still</span> <span m="1503670">have</span> <span m="1503910">to</span> <span m="1504050">look</span> <span m="1504300">at</span> <span m="1504540">the</span> <span m="1504630">intersection</span> <span m="1505080">of</span> <span m="1505150">all</span> <span m="1505410">of</span> <span m="1505500">those</span> <span m="1505790">three</span> <span m="1507230">in</span> <span m="1507420">order</span> <span m="1507580">to</span> <span m="1508110">use</span> <span m="1508470">the</span> <span m="1508640">inclusion-exclusion</span> <span m="1509480">principle.</span> <span m="1511140">And,</span> <span m="1512280">well, the</span> <span m="1512530">same</span> <span m="1512790">trick</span> <span m="1514010">can</span> <span m="1514190">be</span> <span m="1514300">used</span> <span m="1514620">over</span> <span m="1514890">here.</span> </p>
<p><span m="1516640">So</span> <span m="1516960">we</span> <span m="1517110">look</span> <span m="1517330">at</span> <span m="1517450">the</span> <span m="1517540">intersection</span> <span m="1518360">of</span> <span m="1518560">P60</span> <span m="1520030">with</span> <span m="1521070">P04</span> <span m="1522980">and</span> <span m="1523350">the</span> <span m="1523440">permutations</span> <span m="1524240">that</span> <span m="1524330">have</span> <span m="1524520">4 and</span> <span m="1524910">2</span> <span m="1525090">next</span> <span m="1525340">to</span> <span m="1525430">one</span> <span m="1525580">another.</span> <span m="1526400">So</span> <span m="1526450">what</span> <span m="1526540">does</span> <span m="1526740">that</span> <span m="1526850">mean?</span> <span m="1528650">It</span> <span m="1528670">means</span> <span m="1528850">that</span> <span m="1528940">the</span> <span m="1529090">permutation</span> <span m="1529590">that's</span> <span m="1529750">in</span> <span m="1529890">here</span> <span m="1530300">has</span> <span m="1530710">a</span> <span m="1530790">6,</span> <span m="1531290">0</span> <span m="1531910">and</span> <span m="1532090">then</span> <span m="1532260">a</span> <span m="1532310">4</span> <span m="1532620">and</span> <span m="1532700">then</span> <span m="1532840">a</span> <span m="1532920">2.</span> <span m="1533260">So</span> <span m="1534030">these</span> <span m="1534240">are</span> <span m="1534310">all</span> <span m="1534490">the</span> <span m="1534570">permutations</span> <span m="1535860">with</span> <span m="1536050">a</span> <span m="1536190">6,</span> <span m="1536690">0, 4,</span> <span m="1537190">2</span> <span m="1538200">next</span> <span m="1538450">to</span> <span m="1538530">one</span> <span m="1538710">another.</span> </p>
<p><span m="1540160">And</span> <span m="1542910">this</span> <span m="1543120">can</span> <span m="1543560">map</span> <span m="1543920">easily</span> <span m="1545870">to</span> <span m="1545950">the</span> <span m="1546160">sets</span> <span m="1546630">where we</span> <span m="1546880">treat</span> <span m="1547290">this</span> <span m="1547610">as</span> <span m="1547810">one</span> <span m="1548010">symbol--</span> <span m="1548640">6042--</span> <span m="1550040">and</span> <span m="1550180">then</span> <span m="1550320">we</span> <span m="1550480">have 1,</span> <span m="1550770">3,</span> <span m="1551330">5,</span> <span m="1552100">7,</span> <span m="1553280">8,</span> <span m="1553710">and 9.</span> <span m="1555470">This</span> <span m="1555750">set</span> <span m="1556020">has</span> <span m="1556260">seven</span> <span m="1556650">elements,</span> <span m="1557770">so</span> <span m="1558130">the</span> <span m="1558220">intersection</span> <span m="1559680">of</span> <span m="1560570">these</span> <span m="1560980">three</span> <span m="1561720">is</span> <span m="1561920">equal</span> <span m="1562220">to</span> <span m="1562380">7</span> <span m="1562720">factorial.</span> <span m="1564630">And</span> <span m="1564780">now</span> <span m="1564950">we</span> <span m="1565070">can</span> <span m="1565310">use</span> <span m="1565930">the</span> <span m="1566010">inclusion-exclusion</span> <span m="1567190">principle</span> <span m="1568540">using</span> <span m="1569210">this</span> <span m="1569470">over</span> <span m="1569760">here.</span> </p>
<p><span m="1572450">And</span> <span m="1574780">when</span> <span m="1574870">we</span> <span m="1575030">plug</span> <span m="1575360">everything</span> <span m="1575890">in</span> <span m="1576020">here,</span> <span m="1576600">we</span> <span m="1576720">can</span> <span m="1576880">see</span> <span m="1577230">that</span> <span m="1577600">the</span> <span m="1577980">intersection--</span> <span m="1579490">that</span> <span m="1579680">the</span> <span m="1579720">union--</span> <span m="1580750">when</span> <span m="1580990">we</span> <span m="1581480">count</span> <span m="1581770">the</span> <span m="1581930">union of</span> <span m="1582160">P60</span> <span m="1582970">which</span> <span m="1583730">P04</span> <span m="1585230">and</span> <span m="1585440">P42</span> <span m="1587120">is</span> <span m="1587330">actually</span> <span m="1587850">equal</span> <span m="1588240">to</span> <span m="1589130">9</span> <span m="1589400">factorial</span> <span m="1590040">for</span> <span m="1590200">these,</span> <span m="1590920">8</span> <span m="1591260">factorials</span> <span m="1591840">for</span> <span m="1591950">those,</span> <span m="1592480">and</span> <span m="1592630">a</span> <span m="1592680">7</span> <span m="1592970">factorial</span> <span m="1593470">for these.</span> <span m="1593930">So</span> <span m="1594080">it's</span> <span m="1594300">3</span> <span m="1595080">times</span> <span m="1595540">9</span> <span m="1595790">factorial</span> <span m="1597080">minus</span> <span m="1597610">3</span> <span m="1597870">times</span> <span m="1598360">8</span> <span m="1598590">factorial</span> <span m="1599990">plus</span> <span m="1601310">1</span> <span m="1601590">times</span> <span m="1602130">a</span> <span m="1602430">7</span> <span m="1602730">factorial.</span> <span m="1604120">So</span> <span m="1604170">this</span> <span m="1604350">is</span> <span m="1604450">how</span> <span m="1604590">we</span> <span m="1604650">can</span> <span m="1604800">use</span> <span m="1605050">inclusion-exclusion</span> <span m="1605950">principle.</span> </p>
<p><span m="1611900">So</span> <span m="1613250">this</span> <span m="1613580">generalized</span> <span m="1613935">the</span> <span m="1614290">sum</span> <span m="1614570">rule.</span> <span m="1615140">And</span> <span m="1616750">now</span> <span m="1616890">we</span> <span m="1616980">have</span> <span m="1617160">a</span> <span m="1617210">whole</span> <span m="1617410">set</span> <span m="1617720">of</span> <span m="1617880">rules</span> <span m="1618220">already</span> <span m="1618930">discussed</span> <span m="1619510">since</span> <span m="1619980">last</span> <span m="1620230">time.</span> <span m="1620910">We</span> <span m="1621000">continue</span> <span m="1621380">with</span> <span m="1621500">the</span> <span m="1621570">bookkeeper</span> <span m="1622050">rule.</span> <span m="1622360">You</span> <span m="1622670">have already</span> <span m="1622970">seen</span> <span m="1623070">it</span> <span m="1623540">during</span> <span m="1623620">recitation</span> <span m="1624340">so</span> <span m="1624550">I</span> <span m="1624620">will</span> <span m="1624820">only</span> <span m="1625950">write</span> <span m="1626190">it</span> <span m="1626310">out</span> <span m="1626490">once</span> <span m="1626750">more.</span> <span m="1627990">And</span> <span m="1629570">all</span> <span m="1629810">these</span> <span m="1629970">rules</span> <span m="1630270">together</span> <span m="1631080">we</span> <span m="1631250">will</span> <span m="1631360">then</span> <span m="1631570">use</span> <span m="1632040">in</span> <span m="1632850">a</span> <span m="1632970">set</span> <span m="1633160">of</span> <span m="1633280">examples.</span> </p>
<p><span m="1639070">So</span> <span m="1639310">what</span> <span m="1639450">was the</span> <span m="1639890">bookkeeper</span> <span m="1640220">rule?</span> <span m="1641060">We</span> <span m="1641260">have</span> <span m="1643150">that--</span> <span m="1649320">so</span> <span m="1649500">the</span> <span m="1649590">bookkeeper</span> <span m="1650070">rule</span> <span m="1653430">is,</span> <span m="1654600">if</span> <span m="1654830">I</span> <span m="1654950">have</span> <span m="1655700">distinct</span> <span m="1656440">copies</span> <span m="1657330">of</span> <span m="1657520">letters--</span> <span m="1658620">so</span> <span m="1658740">distinct</span> <span m="1659460">copies</span> <span m="1661050">of</span> <span m="1662760">letters</span> <span m="1666800">l1,</span> <span m="1668768">l2,</span> <span m="1670420">l3</span> <span m="1671600">and</span> <span m="1672070">lk,</span> <span m="1674920">well,</span> <span m="1675240">then</span> <span m="1676170">the</span> <span m="1676260">number</span> <span m="1676750">of</span> <span m="1676970">sequences</span> <span m="1681140">that</span> <span m="1681340">have</span> <span m="1681830">exactly</span> <span m="1682560">n1</span> <span m="1683490">letters</span> <span m="1683850">of the</span> <span m="1684140">type</span> <span m="1684430">l1</span> <span m="1685010">and</span> <span m="1685500">n2</span> <span m="1685670">letters</span> <span m="1686060">of</span> <span m="1686180">the</span> <span m="1686290">type</span> <span m="1686530">l2</span> <span m="1687100">and</span> <span m="1687250">so</span> <span m="1687450">on,</span> <span m="1688380">well,</span> <span m="1688490">we</span> <span m="1688610">can</span> <span m="1688830">count</span> <span m="1689120">those.</span> <span m="1690120">So</span> <span m="1690390">sequences</span> <span m="1691060">with</span> <span m="1692750">n1</span> <span m="1693270">copies</span> <span m="1695740">of</span> <span m="1696160">l1</span> <span m="1697970">and</span> <span m="1699020">n2</span> <span m="1700480">copies</span> <span m="1702020">of</span> <span m="1702790">l2,</span> <span m="1704080">and</span> <span m="1704360">then</span> <span m="1705260">we</span> <span m="1705350">continue</span> <span m="1705770">like</span> <span m="1706010">this</span> <span m="1706770">until</span> <span m="1706950">we</span> <span m="1707070">have</span> <span m="1707340">nk</span> <span m="1707880">copies</span> <span m="1709730">of</span> <span m="1710660">lk.</span> </p>
<p><span m="1712200">Well, we</span> <span m="1712310">can</span> <span m="1712510">count</span> <span m="1712770">this.</span> <span m="1713800">So</span> <span m="1714510">these</span> <span m="1714710">copies</span> <span m="1714990">can</span> <span m="1715110">be</span> <span m="1715230">in an</span> <span m="1715370">arbitrary</span> <span m="1715840">order in</span> <span m="1716240">the</span> <span m="1716320">sequence.</span> <span m="1717480">And</span> <span m="1717630">we</span> <span m="1717740">saw</span> <span m="1718200">in</span> <span m="1718330">recitation</span> <span m="1719300">that</span> <span m="1720190">this</span> <span m="1720380">can</span> <span m="1720520">be</span> <span m="1720650">written</span> <span m="1721100">as</span> <span m="1722180">n</span> <span m="1722480">plus</span> <span m="1722820">1</span> <span m="1723140">plus</span> <span m="1724960">n2</span> <span m="1725720">all</span> <span m="1725970">the</span> <span m="1726060">way</span> <span m="1726270">up</span> <span m="1726380">to</span> <span m="1726570">nk</span> <span m="1728660">factorial.</span> <span m="1730350">And</span> <span m="1730540">we divide</span> <span m="1731100">out</span> <span m="1731360">the</span> <span m="1731550">product</span> <span m="1733000">that</span> <span m="1733170">starts</span> <span m="1733550">with</span> <span m="1733750">n1</span> <span m="1734170">factorial</span> <span m="1735440">times</span> <span m="1735810">n2</span> <span m="1736290">factorial</span> <span m="1738850">up</span> <span m="1739040">to</span> <span m="1739410">nk</span> <span m="1740520">factorial.</span> <span m="1742040">And</span> <span m="1742250">this</span> <span m="1743040">we</span> <span m="1743580">also</span> <span m="1745070">can</span> <span m="1745230">write</span> <span m="1745450">as</span> <span m="1745780">what--</span> <span m="1746310">well,</span> <span m="1746470">this</span> <span m="1746670">is actually the</span> <span m="1746960">definition</span> <span m="1747870">of</span> <span m="1748080">what we</span> <span m="1748210">call</span> <span m="1748440">the</span> <span m="1748520">multinomial</span> <span m="1749230">coefficient.</span> </p>
<p><span m="1749900">So</span> <span m="1750445">you've</span> <span m="1750820">already</span> <span m="1751060">seen</span> <span m="1751300">the</span> <span m="1751380">binomial</span> <span m="1752020">coefficients,</span> <span m="1754550">which</span> <span m="1754740">is a</span> <span m="1754820">special</span> <span m="1755290">case</span> <span m="1755580">of</span> <span m="1755700">this</span> <span m="1755940">one.</span> <span m="1756530">So</span> <span m="1756710">we</span> <span m="1756860">write</span> <span m="1757130">this</span> <span m="1757410">as</span> <span m="1759096">n1</span> <span m="1759800">plus</span> <span m="1762060">up</span> <span m="1762240">to</span> <span m="1762480">nk.</span> <span m="1763030">And</span> <span m="1763230">then we</span> <span m="1763370">have</span> <span m="1763550">n1,</span> <span m="1764700">n2,</span> <span m="1765790">and we</span> <span m="1766040">just</span> <span m="1766190">repeat</span> <span m="1766440">all</span> <span m="1766785">of</span> <span m="1767130">those</span> <span m="1768490">in</span> <span m="1768680">here.</span> <span m="1769770">If</span> <span m="1769970">we</span> <span m="1770070">have</span> <span m="1770350">k</span> <span m="1770600">equals</span> <span m="1771060">2,</span> <span m="1772968">then</span> <span m="1773370">we</span> <span m="1773500">get</span> <span m="1773720">the</span> <span m="1773810">binomial</span> <span m="1776470">coefficient.</span> <span m="1778800">And</span> <span m="1780250">if</span> <span m="1780460">k</span> <span m="1780740">equals</span> <span m="1781070">this</span> <span m="1781170">2,</span> <span m="1781820">we</span> <span m="1782060">also</span> <span m="1782440">often</span> <span m="1782750">just</span> <span m="1783000">forget</span> <span m="1783310">about</span> <span m="1783590">the</span> <span m="1783680">last</span> <span m="1784010">term.</span> <span m="1785060">So</span> <span m="1786040">we</span> <span m="1786180">would</span> <span m="1786340">get</span> <span m="1786740">expressions</span> <span m="1787540">that</span> <span m="1787700">look</span> <span m="1787930">like</span> <span m="1788120">this,</span> <span m="1788800">which</span> <span m="1788900">is</span> <span m="1789060">really</span> <span m="1789530">equal</span> <span m="1789710">to</span> <span m="1789940">nk</span> <span m="1791180">comma</span> <span m="1791780">n</span> <span m="1792060">minus</span> <span m="1792470">k.</span> </p>
<p><span m="1794600">OK.</span> <span m="1794890">So</span> <span m="1796020">this</span> <span m="1796240">is</span> <span m="1796320">some--</span> <span m="1797140">these</span> <span m="1797320">are</span> <span m="1797390">some</span> <span m="1797560">definitions.</span> <span m="1799220">And</span> <span m="1801020">we</span> <span m="1801140">can</span> <span m="1801330">apply</span> <span m="1801650">this</span> <span m="1801850">bookkeeper</span> <span m="1802320">rule.</span> <span m="1805480">For</span> <span m="1805700">example,</span> <span m="1806140">last</span> <span m="1806440">time</span> <span m="1806800">in the</span> <span m="1807170">lecture</span> <span m="1807620">we</span> <span m="1807730">were</span> <span m="1808630">talking</span> <span m="1809060">about</span> <span m="1809800">the</span> <span m="1810300">number</span> <span m="1810530">of</span> <span m="1810680">bit</span> <span m="1810900">sequences</span> <span m="1811530">of</span> <span m="1811650">length</span> <span m="1811880">16</span> <span m="1812860">with</span> <span m="1813060">four</span> <span m="1813350">1's.</span> <span m="1815000">We</span> <span m="1815110">wanted</span> <span m="1815290">to</span> <span m="1815480">count this</span> <span m="1815970">because</span> <span m="1816890">we</span> <span m="1817040">found</span> <span m="1817350">out</span> <span m="1817570">that</span> <span m="1818170">if</span> <span m="1818380">you</span> <span m="1818540">want</span> <span m="1818740">to</span> <span m="1819880">select</span> <span m="1821040">12</span> <span m="1821930">donuts</span> <span m="1822620">of</span> <span m="1823210">five</span> <span m="1823610">varieties,</span> <span m="1825660">we</span> <span m="1825820">can</span> <span m="1826220">find</span> <span m="1826490">a</span> <span m="1826550">mapping</span> <span m="1827010">of bijection</span> <span m="1827980">towards</span> <span m="1828690">this</span> <span m="1828870">set</span> <span m="1829270">of</span> <span m="1829520">bit</span> <span m="1829700">sequences</span> <span m="1830470">with</span> <span m="1831000">12</span> <span m="1831320">0's</span> <span m="1831650">and</span> <span m="1831920">four</span> <span m="1832190">1's.</span> </p>
<p><span m="1833380">So</span> <span m="1834800">the</span> <span m="1834900">bookkeeper</span> <span m="1835175">rule</span> <span m="1835450">will</span> <span m="1835770">tell</span> <span m="1835980">us</span> <span m="1836130">exactly</span> <span m="1836580">how</span> <span m="1836700">many</span> <span m="1836940">there</span> <span m="1837140">are.</span> <span m="1839870">And</span> <span m="1840880">that's</span> <span m="1841810">the</span> <span m="1841970">most</span> <span m="1842240">basic</span> <span m="1842600">example,</span> <span m="1843260">essentially,</span> <span m="1844650">of</span> <span m="1844870">this</span> <span m="1846220">way</span> <span m="1846440">of</span> <span m="1846590">counting.</span> <span m="1849300">So</span> <span m="1849870">what</span> <span m="1849990">do</span> <span m="1850090">we</span> <span m="1850220">have?</span> <span m="1853250">So</span> <span m="1853430">the</span> <span m="1853510">number</span> <span m="1853760">of</span> <span m="1853940">bit</span> <span m="1854110">sequences</span> <span m="1857260">of</span> <span m="1857560">length</span> <span m="1860090">16</span> <span m="1861260">and</span> <span m="1862860">with</span> <span m="1864520">four</span> <span m="1864890">1's--</span> <span m="1866320">well,</span> <span m="1866610">is</span> <span m="1866900">exactly</span> <span m="1867480">equal</span> <span m="1867690">to,</span> <span m="1867800">according</span> <span m="1868090">to</span> <span m="1868220">this</span> <span m="1868430">rule,</span> <span m="1869160">as</span> <span m="1871490">length</span> <span m="1871740">16</span> <span m="1872770">and</span> <span m="1873000">we</span> <span m="1873130">need</span> <span m="1873350">to</span> <span m="1873460">choose</span> <span m="1873990">four</span> <span m="1874350">1's</span> <span m="1874930">out</span> <span m="1875140">of</span> <span m="1875310">these</span> <span m="1875440">16.</span> <span m="1875990">And it's</span> <span m="1876240">the</span> <span m="1876290">binomial</span> <span m="1876830">coefficients</span> <span m="1878180">where we</span> <span m="1881930">choose</span> <span m="1882240">four</span> <span m="1882550">out</span> <span m="1882730">of</span> <span m="1882810">the</span> <span m="1882920">16</span> <span m="1883870">symbols.</span> <span m="1886250">And</span> <span m="1886430">this</span> <span m="1886590">is</span> <span m="1886720">equal</span> <span m="1887020">to</span> <span m="1887810">16</span> <span m="1888140">factorial</span> <span m="1889320">divided</span> <span m="1889770">by</span> <span m="1889900">4</span> <span m="1890180">factorial</span> <span m="1890790">times</span> <span m="1891090">12</span> <span m="1891830">factorial.</span> </p>
<p><span m="1895570">Now</span> <span m="1895740">we</span> <span m="1895870">often</span> <span m="1896800">also</span> <span m="1899650">denote</span> <span m="1902100">the</span> <span m="1902200">following</span> <span m="1902650">rule--</span> <span m="1903730">we</span> <span m="1903790">sort</span> <span m="1903990">of</span> <span m="1904110">identify</span> <span m="1904445">as</span> <span m="1904780">a</span> <span m="1904830">special</span> <span m="1905410">case.</span> <span m="1907310">We</span> <span m="1907380">say</span> <span m="1907660">that</span> <span m="1907810">the</span> <span m="1907850">number</span> <span m="1908280">of</span> <span m="1910810">k</span> <span m="1911060">elements--</span> <span m="1912950">of</span> <span m="1913100">k</span> <span m="1913300">element</span> <span m="1913740">subsets--</span> <span m="1918050">of</span> <span m="1918470">an</span> <span m="1919030">n</span> <span m="1919540">element</span> <span m="1922070">set</span> <span m="1923850">is</span> <span m="1924010">actually</span> <span m="1924370">equal</span> <span m="1924760">to</span> <span m="1925432">n</span> <span m="1925770">choose</span> <span m="1926050">k.</span> <span m="1929840">So</span> <span m="1930020">this</span> <span m="1930250">is</span> <span m="1930290">an</span> <span m="1930400">important</span> <span m="1930660">rule</span> <span m="1930920">to</span> <span m="1931050">remember</span> <span m="1932120">which</span> <span m="1932270">will</span> <span m="1932460">occur</span> <span m="1933080">many</span> <span m="1933350">times.</span> </p>
<p><span m="1942370">OK.</span> <span m="1943780">A</span> <span m="1943890">theorem</span> <span m="1944350">that</span> <span m="1944550">we</span> <span m="1944630">can</span> <span m="1944810">have</span> <span m="1945150">derived</span> <span m="1945580">from</span> <span m="1945650">all</span> <span m="1945790">this</span> <span m="1945940">stuff</span> <span m="1946590">is</span> <span m="1946860">what</span> <span m="1947000">we</span> <span m="1947110">call</span> <span m="1947480">the</span> <span m="1948360">binomial</span> <span m="1948900">theorem.</span> <span m="1949510">So</span> <span m="1949910">we'll quickly</span> <span m="1950220">go</span> <span m="1950540">over</span> <span m="1950680">that.</span> <span m="1957090">The</span> <span m="1957500">binomial</span> <span m="1957990">theorem</span> <span m="1958450">says</span> <span m="1958930">that,</span> <span m="1959880">for</span> <span m="1960210">all</span> <span m="1960780">integers</span> <span m="1961350">n--</span> <span m="1961580">positive</span> <span m="1961840">integers</span> <span m="1962310">n--</span> <span m="1963430">we</span> <span m="1963530">have</span> <span m="1963980">a</span> <span m="1964220">plus</span> <span m="1964640">b</span> <span m="1964980">to</span> <span m="1965120">the</span> <span m="1965270">power</span> <span m="1965610">n</span> <span m="1966420">equals</span> <span m="1966840">to</span> <span m="1967010">the</span> <span m="1967100">sum</span> <span m="1968480">where we</span> <span m="1969160">take</span> <span m="1969680">k</span> <span m="1970050">from</span> <span m="1970320">0</span> <span m="1970740">to</span> <span m="1971060">n</span> <span m="1972320">and</span> <span m="1972620">then</span> <span m="1974730">we</span> <span m="1974870">have</span> <span m="1975030">the</span> <span m="1975120">binomial</span> <span m="1976420">where we</span> <span m="1977250">have</span> <span m="1977570">this</span> <span m="1977740">expression</span> <span m="1978230">over</span> <span m="1978480">here.</span> <span m="1978900">We</span> <span m="1979330">choose</span> <span m="1979650">k</span> <span m="1980240">out</span> <span m="1980460">of</span> <span m="1980550">n</span> <span m="1981470">and</span> <span m="1983590">we</span> <span m="1983700">choose,</span> <span m="1983990">say,</span> <span m="1984870">k</span> <span m="1986840">times</span> <span m="1987640">b</span> <span m="1988020">in</span> <span m="1988130">this</span> <span m="1988300">expression.</span> <span m="1989010">I</span> <span m="1989090">will</span> <span m="1989210">explain</span> <span m="1989630">in a</span> <span m="1989780">moment</span> <span m="1989960">by</span> <span m="1990080">an</span> <span m="1990160">example.</span> <span m="1990690">And</span> <span m="1990810">n</span> <span m="1990940">minus</span> <span m="1991300">k</span> <span m="1991620">ace.</span> </p>
<p><span m="1993630">And</span> <span m="1993800">this</span> <span m="1993960">is</span> <span m="1994130">the</span> <span m="1994660">theorem.</span> <span m="1995170">And</span> <span m="1995340">I</span> <span m="1995360">will</span> <span m="1995540">just</span> <span m="1995750">give an</span> <span m="1996080">example</span> <span m="1996750">just</span> <span m="1997300">to</span> <span m="1997430">show</span> <span m="1998230">how</span> <span m="1998700">we</span> <span m="1998790">can</span> <span m="1998950">think</span> <span m="1999200">about</span> <span m="1999540">this.</span> <span m="2002300">So</span> <span m="2002470">let me do that</span> <span m="2002950">over</span> <span m="2003190">here.</span> <span m="2006050">So</span> <span m="2006170">if</span> <span m="2006290">you</span> <span m="2006380">take</span> <span m="2006990">n</span> <span m="2007220">equals</span> <span m="2007750">to</span> <span m="2007960">2,</span> <span m="2008650">then</span> <span m="2011410">we can</span> <span m="2011780">see that</span> <span m="2011870">we</span> <span m="2012060">get</span> <span m="2012290">all</span> <span m="2012660">the</span> <span m="2012770">combinations,</span> <span m="2014270">essentially,</span> <span m="2014830">of</span> <span m="2014920">a</span> <span m="2015080">and</span> <span m="2015490">b</span> <span m="2015760">of length</span> <span m="2016160">2.</span> <span m="2016380">So</span> <span m="2016570">you</span> <span m="2016710">have</span> <span m="2017250">a</span> <span m="2017410">times</span> <span m="2017860">a,</span> <span m="2018200">a</span> <span m="2018540">times</span> <span m="2018900">b,</span> <span m="2019230">b</span> <span m="2019560">times</span> <span m="2019930">a, and</span> <span m="2020330">b</span> <span m="2020490">times</span> <span m="2020820">b.</span> </p>
<p><span m="2022420">So</span> <span m="2023500">what</span> <span m="2023640">do</span> <span m="2023740">we</span> <span m="2023870">see?</span> <span m="2024300">We</span> <span m="2024310">see</span> <span m="2025020">a</span> <span m="2025280">squared</span> <span m="2026190">plus</span> <span m="2026730">ab</span> <span m="2028060">plus</span> <span m="2028360">ba</span> <span m="2029180">plus</span> <span m="2029790">b</span> <span m="2030120">squared.</span> <span m="2031470">Now</span> <span m="2031510">we</span> <span m="2031610">see</span> <span m="2031820">that</span> <span m="2032050">these</span> <span m="2032690">have</span> <span m="2033010">1</span> <span m="2033280">times</span> <span m="2033750">an a</span> <span m="2034502">and</span> <span m="2034880">1</span> <span m="2035150">times</span> <span m="2035445">a</span> <span m="2035740">b.</span> <span m="2036980">This</span> <span m="2037180">one</span> <span m="2037400">has</span> <span m="2037810">2</span> <span m="2038010">times</span> <span m="2038350">an</span> <span m="2038690">a</span> <span m="2039185">and</span> <span m="2039680">this</span> <span m="2039870">one</span> <span m="2040060">has</span> <span m="2040530">2</span> <span m="2040750">times</span> <span m="2041070">a</span> <span m="2041390">b.</span> <span m="2044420">And</span> <span m="2044610">this</span> <span m="2044800">combines</span> <span m="2045240">together</span> <span m="2045980">as</span> <span m="2046690">a</span> <span m="2046870">squared</span> <span m="2047410">plus</span> <span m="2047720">2</span> <span m="2047940">times</span> <span m="2048449">ab</span> <span m="2049250">plus</span> <span m="2049730">b</span> <span m="2049940">squared.</span> </p>
<p><span m="2051340">Now,</span> <span m="2051380">if</span> <span m="2051530">you</span> <span m="2051639">have</span> <span m="2051860">n</span> <span m="2052090">equals</span> <span m="2052520">3--</span> <span m="2053739">so</span> <span m="2053969">we</span> <span m="2054090">have</span> <span m="2055190">a</span> <span m="2055310">plus b</span> <span m="2055679">to</span> <span m="2055790">the</span> <span m="2055889">power</span> <span m="2056179">3--</span> <span m="2056989">you</span> <span m="2057100">will</span> <span m="2057300">get</span> <span m="2057949">all</span> <span m="2058199">the</span> <span m="2058280">different</span> <span m="2061550">products,</span> <span m="2062100">all</span> <span m="2062210">the</span> <span m="2062280">different</span> <span m="2062670">terms,</span> <span m="2063179">that</span> <span m="2063810">add</span> <span m="2064040">up</span> <span m="2064620">to</span> <span m="2065330">this,</span> <span m="2066889">are</span> <span m="2067489">all</span> <span m="2067670">the</span> <span m="2067750">kinds of</span> <span m="2068010">combinations</span> <span m="2068600">of</span> <span m="2068750">a's</span> <span m="2069030">and</span> <span m="2069150">b's.</span> <span m="2069940">So</span> <span m="2070210">for</span> <span m="2070370">example,</span> <span m="2071770">we</span> <span m="2071870">have</span> <span m="2072460">three</span> <span m="2072880">a's--</span> <span m="2074909">so</span> <span m="2075100">3</span> <span m="2075340">times</span> <span m="2075790">a--</span> <span m="2076489">but</span> <span m="2076719">it can</span> <span m="2076880">also</span> <span m="2077250">have</span> <span m="2080130">a</span> <span m="2080320">squared</span> <span m="2080770">times</span> <span m="2081139">b</span> <span m="2081790">plus</span> <span m="2082900">aba</span> <span m="2084580">plus</span> <span m="2085480">b</span> <span m="2086290">a</span> <span m="2086360">squared.</span> <span m="2087739">And</span> <span m="2088350">those</span> <span m="2089090">have</span> <span m="2089360">exactly</span> <span m="2089850">two</span> <span m="2090040">times</span> <span m="2090530">an</span> <span m="2090884">a and</span> <span m="2091239">one</span> <span m="2091440">times</span> <span m="2091730">a b.</span> </p>
<p><span m="2093840">I</span> <span m="2093980">can</span> <span m="2094360">now</span> <span m="2094560">look</span> <span m="2094780">at</span> <span m="2095920">all</span> <span m="2096120">the</span> <span m="2096250">others--</span> <span m="2097650">so</span> <span m="2097790">I</span> <span m="2097940">may</span> <span m="2098390">have</span> <span m="2098680">two</span> <span m="2098860">times</span> <span m="2099160">the</span> <span m="2099250">b</span> <span m="2099370">and</span> <span m="2099470">one</span> <span m="2099630">times</span> <span m="2100030">the a.</span> <span m="2100600">So</span> <span m="2101350">what</span> <span m="2101530">our</span> <span m="2101640">those?</span> <span m="2102570">We</span> <span m="2102690">have</span> <span m="2103010">a</span> <span m="2103190">times</span> <span m="2103580">b</span> <span m="2103770">squared</span> <span m="2105430">plus</span> <span m="2106310">bab</span> <span m="2108210">plus</span> <span m="2109430">b</span> <span m="2109640">squared</span> <span m="2110490">a.</span> <span m="2111570">So</span> <span m="2111910">I</span> <span m="2112000">count</span> <span m="2113130">these</span> <span m="2115860">as</span> <span m="2115990">the</span> <span m="2116080">same,</span> <span m="2116360">right?</span> <span m="2116550">They</span> <span m="2116700">each</span> <span m="2117100">have</span> <span m="2117330">the</span> <span m="2117400">same</span> <span m="2117820">number</span> <span m="2118110">of</span> <span m="2118780">b's.</span> <span m="2119340">Two</span> <span m="2119490">times</span> <span m="2119800">the</span> <span m="2119880">b</span> <span m="2120410">and</span> <span m="2120590">one</span> <span m="2120780">times</span> <span m="2121090">the a.</span> <span m="2122510">And then,</span> <span m="2122540">finally,</span> <span m="2124740">I</span> <span m="2124960">have</span> <span m="2125570">b</span> <span m="2125880">to</span> <span m="2126000">the</span> <span m="2126090">power</span> <span m="2126370">3</span> <span m="2127110">which</span> <span m="2127320">counts</span> <span m="2128020">three</span> <span m="2128300">times</span> <span m="2128620">the</span> <span m="2128700">b.</span> </p>
<p><span m="2129180">So</span> <span m="2129320">these</span> <span m="2129550">are all</span> <span m="2129770">the</span> <span m="2129860">possible</span> <span m="2130300">terms</span> <span m="2130580">that</span> <span m="2130690">I have.</span> <span m="2131120">There</span> <span m="2131270">are</span> <span m="2131670">eight in</span> <span m="2132070">total.</span> <span m="2132510">One,</span> <span m="2132800">two,</span> <span m="2132960">three,</span> <span m="2133210">four,</span> <span m="2133510">five,</span> <span m="2133840">up</span> <span m="2134170">to</span> <span m="2134330">eight.</span> <span m="2135510">And</span> <span m="2135690">that</span> <span m="2135790">fits,</span> <span m="2136100">right?</span> <span m="2136310">Because</span> <span m="2136650">I</span> <span m="2136740">can</span> <span m="2136970">have</span> <span m="2137190">a</span> <span m="2137300">choice</span> <span m="2138390">of an</span> <span m="2138720">a and</span> <span m="2139150">a b.</span> <span m="2139890">And I</span> <span m="2139960">do</span> <span m="2140090">that</span> <span m="2140200">three</span> <span m="2140480">times.</span> <span m="2140960">So 2</span> <span m="2141180">times</span> <span m="2141520">2</span> <span m="2141680">times</span> <span m="2141980">2</span> <span m="2142110">is</span> <span m="2142230">equal</span> <span m="2142450">to</span> <span m="2142590">8.</span> <span m="2143440">So</span> <span m="2143730">I have eight</span> <span m="2144080">terms.</span> </p>
<p><span m="2145090">And</span> <span m="2145240">I</span> <span m="2145280">can</span> <span m="2145460">group</span> <span m="2145720">them</span> <span m="2145890">together</span> <span m="2147960">by</span> <span m="2148300">taking</span> <span m="2148820">a</span> <span m="2148950">to</span> <span m="2149070">the</span> <span m="2149180">power</span> <span m="2149460">3</span> <span m="2150170">plus</span> <span m="2150840">3</span> <span m="2151280">times</span> <span m="2153150">a</span> <span m="2153350">squared</span> <span m="2153900">b</span> <span m="2154700">plus</span> <span m="2155280">3</span> <span m="2155540">times--</span> <span m="2156730">and</span> <span m="2156890">now</span> <span m="2157080">I</span> <span m="2157180">have</span> <span m="2157430">b</span> <span m="2157610">squared</span> <span m="2158020">a--</span> <span m="2160400">and</span> <span m="2160880">finally,</span> <span m="2161390">b</span> <span m="2161860">to</span> <span m="2162240">the</span> <span m="2162330">power</span> <span m="2162570">3.</span> <span m="2164830">So</span> <span m="2165610">what</span> <span m="2165740">do</span> <span m="2165830">we</span> <span m="2165950">see</span> <span m="2166300">here?</span> <span m="2166980">How</span> <span m="2167150">many</span> <span m="2167500">terms</span> <span m="2167910">are</span> <span m="2168180">their</span> <span m="2168400">that</span> <span m="2168610">have</span> <span m="2168790">exactly</span> <span m="2169260">two</span> <span m="2169420">times</span> <span m="2169860">an a</span> <span m="2170170">and one</span> <span m="2170480">times</span> <span m="2170770">a</span> <span m="2170870">b?</span> <span m="2171630">Well, it's</span> <span m="2171930">like</span> <span m="2172270">we</span> <span m="2172610">have</span> <span m="2172800">a</span> <span m="2172910">set</span> <span m="2173340">of</span> <span m="2174330">three</span> <span m="2175040">elements,</span> <span m="2175990">right?</span> <span m="2177370">Or</span> <span m="2177870">three</span> <span m="2179300">positions.</span> <span m="2180470">And</span> <span m="2180750">I</span> <span m="2180860">like</span> <span m="2181210">two</span> <span m="2181460">of</span> <span m="2181590">them</span> <span m="2181750">to</span> <span m="2181860">be</span> <span m="2182030">an a.  So I</span> <span m="2182470">choose</span> <span m="2182840">two</span> <span m="2183120">out</span> <span m="2183340">of</span> <span m="2183460">those</span> <span m="2183710">three</span> <span m="2183920">to</span> <span m="2184040">be</span> <span m="2184320">an a.</span> <span m="2184580">So</span> <span m="2184770">I use</span> <span m="2185120">this</span> <span m="2185410">subset</span> <span m="2186740">rule</span> <span m="2187960">to</span> <span m="2188100">count</span> <span m="2189180">how</span> <span m="2189400">many</span> <span m="2190300">times</span> <span m="2190750">I</span> <span m="2190870">see</span> <span m="2192000">the</span> <span m="2192110">same</span> <span m="2193130">term</span> <span m="2193700">back.</span> </p>
<p><span m="2195270">So</span> <span m="2198150">let's</span> <span m="2198360">write</span> <span m="2198550">this</span> <span m="2198780">out</span> <span m="2199080">so</span> <span m="2199290">that</span> <span m="2199490">is--</span> <span m="2202570">so</span> <span m="2202650">the</span> <span m="2202760">number</span> <span m="2203090">of</span> <span m="2203240">terms</span> <span m="2205910">that</span> <span m="2206110">have</span> <span m="2208810">k</span> <span m="2209060">times</span> <span m="2210030">an a,</span> <span m="2210960">and</span> <span m="2211890">we</span> <span m="2212060">have,</span> <span m="2212940">say,</span> <span m="2213290">n</span> <span m="2213720">minus</span> <span m="2214090">k</span> <span m="2214360">times</span> <span m="2215360">a</span> <span m="2215720">b.</span> <span m="2216120">So</span> <span m="2216520">how</span> <span m="2216710">many</span> <span m="2216930">terms</span> <span m="2217260">do</span> <span m="2217390">I</span> <span m="2217490">have?</span> <span m="2218680">Well,</span> <span m="2218750">that is</span> <span m="2219210">the</span> <span m="2220310">length</span> <span m="2223910">of--</span> <span m="2224630">so</span> <span m="2224820">the</span> <span m="2224920">number</span> <span m="2225440">of--</span> <span m="2227920">oops.</span> <span m="2229700">Change</span> <span m="2230110">that.</span> <span m="2230280">So</span> <span m="2230410">it's</span> <span m="2230580">the</span> <span m="2230660">number</span> <span m="2231640">of</span> <span m="2231940">length</span> <span m="2232760">n</span> <span m="2233100">sequences</span> <span m="2236320">that</span> <span m="2236620">have</span> <span m="2236810">the</span> <span m="2236900">following</span> <span m="2237240">property.</span> <span m="2238350">That</span> <span m="2238540">these</span> <span m="2238840">have</span> <span m="2240060">k</span> <span m="2240954">a's</span> <span m="2243220">and</span> <span m="2243570">n</span> <span m="2243770">minus</span> <span m="2244220">k</span> <span m="2244920">b's.</span> </p>
<p><span m="2247220">And</span> <span m="2247630">that's</span> <span m="2247820">easy</span> <span m="2248150">to</span> <span m="2248290">count.</span> <span m="2249390">Which</span> <span m="2249620">is</span> <span m="2249740">equal</span> <span m="2250010">to</span> <span m="2250140">what</span> <span m="2250320">we</span> <span m="2250410">have</span> <span m="2250540">done</span> <span m="2250710">here</span> <span m="2250920">before.</span> <span m="2252120">We</span> <span m="2252250">have</span> <span m="2252390">the</span> <span m="2252460">binomial</span> <span m="2254490">n</span> <span m="2254670">choose</span> <span m="2254940">k.</span> <span m="2256700">And</span> <span m="2256930">that,</span> <span m="2257640">if</span> <span m="2257780">you</span> <span m="2257890">plug</span> <span m="2258160">that</span> <span m="2258280">in</span> <span m="2258440">over</span> <span m="2258720">here,</span> <span m="2259130">then</span> <span m="2259260">we</span> <span m="2259350">get</span> <span m="2259580">this</span> <span m="2259850">particular</span> <span m="2260320">sum.</span> <span m="2262400">So</span> <span m="2262560">it</span> <span m="2262700">is--</span> <span m="2263440">this</span> <span m="2263630">is</span> <span m="2263830">the</span> <span m="2263930">binomial</span> <span m="2264330">theorem.</span> <span m="2264980">Very</span> <span m="2265560">famous.</span> </p>
<p><span m="2266910">And</span> <span m="2268450">now</span> <span m="2268580">we're</span> <span m="2268780">going to</span> <span m="2269030">use</span> <span m="2269270">all</span> <span m="2269400">this</span> <span m="2269550">stuff</span> <span m="2269960">to</span> <span m="2270610">count some</span> <span m="2271020">poker</span> <span m="2271450">hands.</span> <span m="2271660">So</span> <span m="2272210">we</span> <span m="2272290">have</span> <span m="2272450">come</span> <span m="2273520">to</span> <span m="2273630">this</span> <span m="2273900">part</span> <span m="2274430">of</span> <span m="2274850">the</span> <span m="2274980">lecture.</span> <span m="2275440">So</span> <span m="2275650">we're</span> <span m="2275720">going</span> <span m="2275990">to do</span> <span m="2276100">a</span> <span m="2276150">bunch</span> <span m="2276390">of</span> <span m="2276490">examples</span> <span m="2277740">that</span> <span m="2277830">are</span> <span m="2277940">similar</span> <span m="2278630">to</span> <span m="2278800">some</span> <span m="2279030">of</span> <span m="2279140">those</span> <span m="2279830">that</span> <span m="2279980">you will</span> <span m="2280150">see</span> <span m="2280330">in the</span> <span m="2280420">problem</span> <span m="2280740">set.</span> <span m="2281460">And</span> <span m="2281730">then at</span> <span m="2281830">the</span> <span m="2281960">end</span> <span m="2282090">we go</span> <span m="2282690">and</span> <span m="2282830">continue</span> <span m="2283145">with</span> <span m="2283460">some--</span> <span m="2284360">with</span> <span m="2284500">a</span> <span m="2284540">different</span> <span m="2284810">proof</span> <span m="2285070">technique.</span> </p>
<p><span m="2287190">So</span> <span m="2292100">let's</span> <span m="2293260">first</span> <span m="2293570">define</span> <span m="2295390">a</span> <span m="2295470">deck</span> <span m="2295680">of</span> <span m="2295810">cards</span> <span m="2296260">because</span> <span m="2296560">that's</span> <span m="2296730">what</span> <span m="2296850">we are</span> <span m="2297070">going</span> <span m="2297270">to</span> <span m="2297370">use</span> <span m="2297690">now.</span> <span m="2298870">So</span> <span m="2299530">for</span> <span m="2300290">this</span> <span m="2300490">type</span> <span m="2300720">of</span> <span m="2300870">problems,</span> <span m="2306130">we</span> <span m="2306300">will</span> <span m="2306430">use</span> <span m="2307150">a</span> <span m="2307700">deck</span> <span m="2308270">which</span> <span m="2308520">is</span> <span m="2310380">actually</span> <span m="2311330">a</span> <span m="2311430">set</span> <span m="2311750">of</span> <span m="2311860">52</span> <span m="2312570">cards.</span> <span m="2314810">And</span> <span m="2318030">a</span> <span m="2318160">card</span> <span m="2318450">itself</span> <span m="2320010">actually</span> <span m="2321020">has</span> <span m="2321420">a</span> <span m="2322060">suit.</span> <span m="2324490">And</span> <span m="2324660">the</span> <span m="2324740">suit</span> <span m="2325100">can</span> <span m="2325350">be</span> <span m="2326490">something</span> <span m="2326980">that</span> <span m="2327140">looks</span> <span m="2327410">like</span> <span m="2327700">this,</span> <span m="2328410">which</span> <span m="2328590">is</span> <span m="2328780">called</span> <span m="2329760">spades.</span> <span m="2331170">S</span> <span m="2331410">of</span> <span m="2331500">spades.</span> <span m="2332970">We</span> <span m="2333120">have</span> <span m="2333290">another</span> <span m="2334120">symbol</span> <span m="2334680">which</span> <span m="2334960">is</span> <span m="2335340">this,</span> <span m="2336090">hearts.</span> <span m="2338500">And</span> <span m="2339150">we</span> <span m="2339290">will</span> <span m="2339480">have</span> <span m="2342530">clubs</span> <span m="2343560">and we</span> <span m="2343860">have</span> <span m="2345170">diamonds.</span> </p>
<p><span m="2346670">And</span> <span m="2346840">besides</span> <span m="2347950">a</span> <span m="2348030">suit,</span> <span m="2348530">a</span> <span m="2348600">card</span> <span m="2348870">also</span> <span m="2349160">has</span> <span m="2349370">a</span> <span m="2349420">value.</span> <span m="2350970">And</span> <span m="2351180">the</span> <span m="2351280">values</span> <span m="2352730">are</span> <span m="2353970">arranged</span> <span m="2354400">from</span> <span m="2354750">2,</span> <span m="2355940">3,</span> <span m="2357120">4,</span> <span m="2358320">all</span> <span m="2358580">the</span> <span m="2358670">way</span> <span m="2358880">up</span> <span m="2359070">to</span> <span m="2359420">10.</span> <span m="2361030">Then</span> <span m="2361230">we</span> <span m="2361380">have</span> <span m="2361860">a</span> <span m="2361950">special</span> <span m="2362750">symbol--</span> <span m="2363560">special</span> <span m="2363950">value--</span> <span m="2364440">which</span> <span m="2364740">is</span> <span m="2364900">called</span> <span m="2365240">the</span> <span m="2365340">Jack.</span> <span m="2366600">We've</span> <span m="2366740">got</span> <span m="2366900">a</span> <span m="2366960">Queen</span> <span m="2367890">and a</span> <span m="2368190">King</span> <span m="2368840">and</span> <span m="2369000">finally</span> <span m="2369440">the</span> <span m="2369660">Ace.</span> <span m="2371030">And</span> <span m="2371240">in</span> <span m="2371430">total,</span> <span m="2372460">we</span> <span m="2372600">have</span> <span m="2372970">13</span> <span m="2373500">possible</span> <span m="2373970">values.</span> <span m="2375930">So</span> <span m="2376200">you</span> <span m="2376310">can</span> <span m="2376470">see</span> <span m="2376820">that</span> <span m="2377110">we</span> <span m="2377240">have</span> <span m="2378000">four</span> <span m="2378410">times,</span> <span m="2378930">because</span> <span m="2379190">of</span> <span m="2379300">four</span> <span m="2379550">different</span> <span m="2379850">suits,</span> <span m="2380530">4</span> <span m="2380890">times</span> <span m="2381260">13</span> <span m="2381900">is</span> <span m="2382130">52</span> <span m="2382760">different</span> <span m="2383300">cards</span> <span m="2384110">and</span> <span m="2384300">that</span> <span m="2384410">makes</span> <span m="2384680">up</span> <span m="2384760">a</span> <span m="2384860">deck.</span> </p>
<p><span m="2386330">So</span> <span m="2386540">we're</span> <span m="2386670">going</span> <span m="2386920">to</span> <span m="2387100">look</span> <span m="2387360">at</span> <span m="2388030">hands.</span> <span m="2389450">And</span> <span m="2391000">a</span> <span m="2391050">hand</span> <span m="2393050">is</span> <span m="2393320">actually</span> <span m="2393770">a</span> <span m="2394140">collection--</span> <span m="2394990">a</span> <span m="2395350">set</span> <span m="2395605">of</span> <span m="2395860">five</span> <span m="2396260">cards.</span> <span m="2397720">So</span> <span m="2397960">it's</span> <span m="2398080">a</span> <span m="2398130">subset</span> <span m="2399540">of</span> <span m="2399770">the</span> <span m="2399880">deck</span> <span m="2402340">of</span> <span m="2402530">five</span> <span m="2403000">cards.</span> <span m="2404560">And</span> <span m="2407560">we</span> <span m="2407680">do</span> <span m="2407810">not</span> <span m="2408170">worry</span> <span m="2408510">about</span> <span m="2408820">the</span> <span m="2408960">order</span> <span m="2409310">of</span> <span m="2409430">the</span> <span m="2409550">cards</span> <span m="2409880">in</span> <span m="2409950">your</span> <span m="2410130">hand.</span> <span m="2411290">They</span> <span m="2411310">can</span> <span m="2411520">be</span> <span m="2412010">permuted,</span> <span m="2412610">if</span> <span m="2412740">you</span> <span m="2412890">want.</span> <span m="2413210">So</span> <span m="2413650">the</span> <span m="2413830">order's</span> <span m="2414085">not</span> <span m="2414340">important.</span> <span m="2415480">So</span> <span m="2415660">it's</span> <span m="2415960">a</span> <span m="2416030">subset</span> <span m="2416680">of</span> <span m="2416780">five</span> <span m="2417220">cards.</span> </p>
<p><span m="2419750">So</span> <span m="2419920">how</span> <span m="2420080">many</span> <span m="2420360">hands</span> <span m="2420710">do</span> <span m="2420860">I</span> <span m="2420980">have?</span> <span m="2424130">Well,</span> <span m="2424820">how</span> <span m="2425030">many</span> <span m="2425290">ways</span> <span m="2425600">are</span> <span m="2425660">there</span> <span m="2425900">to</span> <span m="2426040">choose</span> <span m="2426550">a</span> <span m="2426680">subset</span> <span m="2428000">of</span> <span m="2428080">five</span> <span m="2428510">cards</span> <span m="2428910">out</span> <span m="2429140">of</span> <span m="2429260">a</span> <span m="2429330">deck</span> <span m="2429730">of</span> <span m="2429840">52</span> <span m="2430410">cards?</span> <span m="2431740">Well,</span> <span m="2432230">we</span> <span m="2432350">can</span> <span m="2432490">use</span> <span m="2432670">a</span> <span m="2432730">subset</span> <span m="2433170">through</span> <span m="2433900">and</span> <span m="2434100">we</span> <span m="2434280">get</span> <span m="2434370">the</span> <span m="2434460">binomial</span> <span m="2435020">52</span> <span m="2435700">choose</span> <span m="2437270">5.</span> <span m="2441150">Which</span> <span m="2441380">is,</span> <span m="2441890">by</span> <span m="2442040">the way,</span> <span m="2442630">a</span> <span m="2442730">lot</span> <span m="2443250">so</span> <span m="2443810">let</span> <span m="2443930">me</span> <span m="2444040">write</span> <span m="2444250">it</span> <span m="2444380">out</span> <span m="2444540">as</span> <span m="2444680">well.</span> <span m="2444900">It's</span> <span m="2445050">about</span> <span m="2446320">2</span> <span m="2446600">and</span> <span m="2446870">1/2</span> <span m="2447220">million.</span> </p>
<p><span m="2453430">So</span> <span m="2453650">we</span> <span m="2453800">are</span> <span m="2453840">interested</span> <span m="2454590">in,</span> <span m="2455030">when</span> <span m="2455160">we</span> <span m="2455260">play</span> <span m="2455480">poker--</span> <span m="2457280">if</span> <span m="2457780">you</span> <span m="2457830">do</span> <span m="2457950">that--</span> <span m="2459660">in</span> <span m="2460290">really</span> <span m="2460530">good</span> <span m="2460750">hands.</span> <span m="2461450">So</span> <span m="2461720">we</span> <span m="2461860">like</span> <span m="2462510">four</span> <span m="2462840">of a</span> <span m="2463060">kind,</span> <span m="2463470">which</span> <span m="2463670">means</span> <span m="2463920">that we</span> <span m="2464230">have</span> <span m="2464460">four</span> <span m="2466050">cards</span> <span m="2466580">that</span> <span m="2467180">actually</span> <span m="2467660">have</span> <span m="2468060">the</span> <span m="2468150">same</span> <span m="2468440">value.</span> <span m="2469420">Or</span> <span m="2469840">we</span> <span m="2469990">have</span> <span m="2470630">cards</span> <span m="2470950">that--</span> <span m="2473080">hands</span> <span m="2473510">that</span> <span m="2473730">have</span> <span m="2474160">what</span> <span m="2474380">we</span> <span m="2474450">call a</span> <span m="2474730">full</span> <span m="2474980">house.</span> <span m="2475270">We will</span> <span m="2475510">count</span> <span m="2475770">those</span> <span m="2476090">in</span> <span m="2476170">a</span> <span m="2476220">moment.</span> <span m="2477120">Or</span> <span m="2478190">other</span> <span m="2478380">kinds</span> <span m="2478620">of</span> <span m="2478680">combinations.</span> </p>
<p><span m="2480240">And</span> <span m="2481300">the</span> <span m="2481470">rarer</span> <span m="2482240">the</span> <span m="2482390">combination</span> <span m="2483030">is</span> <span m="2483190">in</span> <span m="2483310">your</span> <span m="2483440">hand,</span> <span m="2485410">the</span> <span m="2485600">higher</span> <span m="2486170">or the</span> <span m="2486420">better</span> <span m="2486900">your</span> <span m="2487150">hand</span> <span m="2487460">is,</span> <span m="2488220">and</span> <span m="2488350">the</span> <span m="2488410">more</span> <span m="2488560">likely</span> <span m="2489820">it</span> <span m="2489970">is</span> <span m="2490120">that you</span> <span m="2490445">win</span> <span m="2491470">the</span> <span m="2491570">poker</span> <span m="2491870">game.</span> <span m="2493080">If</span> <span m="2493330">you</span> <span m="2493420">do</span> <span m="2493500">not</span> <span m="2493680">get</span> <span m="2493840">left</span> <span m="2494110">out.</span> <span m="2495690">All</span> <span m="2495850">right.</span> </p>
<p><span m="2501000">So</span> <span m="2501150">let's</span> <span m="2501900">give</span> <span m="2502110">an</span> <span m="2502220">example</span> <span m="2505070">of</span> <span m="2506760">a</span> <span m="2506920">four</span> <span m="2507210">of</span> <span m="2507320">a</span> <span m="2507420">kind.</span> <span m="2508030">So</span> <span m="2508500">we're</span> <span m="2508660">going</span> <span m="2508860">to</span> <span m="2509060">compute</span> <span m="2509400">those</span> <span m="2509530">and</span> <span m="2509760">we</span> <span m="2509840">will</span> <span m="2509940">see</span> <span m="2510150">that</span> <span m="2510310">we</span> <span m="2510390">need</span> <span m="2510600">to</span> <span m="2510730">do--</span> <span m="2511090">to</span> <span m="2511240">use</span> <span m="2511420">all</span> <span m="2511570">the--</span> <span m="2511780">a</span> <span m="2511940">combination</span> <span m="2512350">for</span> <span m="2512440">all</span> <span m="2512590">these</span> <span m="2512780">rules.</span> <span m="2522260">OK.</span> <span m="2524600">So</span> <span m="2527010">four</span> <span m="2527260">of a</span> <span m="2527550">kind</span> <span m="2529890">is</span> <span m="2530130">the</span> <span m="2530180">special</span> <span m="2530650">hand</span> <span m="2531180">where</span> <span m="2531500">we</span> <span m="2532880">have</span> <span m="2534720">four</span> <span m="2536350">of</span> <span m="2536780">one</span> <span m="2537200">kind</span> <span m="2538260">of</span> <span m="2538750">value.</span> </p>
<p><span m="2543210">Notice,</span> <span m="2543560">by</span> <span m="2543710">the</span> <span m="2543820">way,</span> <span m="2544020">that</span> <span m="2544110">the</span> <span m="2544250">fifth</span> <span m="2544590">card,</span> <span m="2545380">because</span> <span m="2545780">of</span> <span m="2545880">this,</span> <span m="2546200">must</span> <span m="2546470">have</span> <span m="2546830">a</span> <span m="2547420">different</span> <span m="2547770">value,</span> <span m="2548140">right?</span> <span m="2548440">Because</span> <span m="2549950">there</span> <span m="2550050">are</span> <span m="2550320">only</span> <span m="2550730">four</span> <span m="2551180">cards</span> <span m="2551780">that</span> <span m="2551970">have,</span> <span m="2552290">say,</span> <span m="2552590">the</span> <span m="2552770">value</span> <span m="2553190">eight.</span> <span m="2554090">Because</span> <span m="2555170">there</span> <span m="2555270">are</span> <span m="2555350">only</span> <span m="2555750">four</span> <span m="2556020">different</span> <span m="2556360">suits.</span> <span m="2557030">So</span> <span m="2557210">the</span> <span m="2557290">fifth</span> <span m="2557550">card</span> <span m="2557740">will</span> <span m="2557950">ultimately</span> <span m="2558540">have</span> <span m="2558770">a</span> <span m="2558830">different</span> <span m="2559220">value.</span> </p>
<p><span m="2561650">So</span> <span m="2561840">as</span> <span m="2561970">an</span> <span m="2562030">example,</span> <span m="2564580">we</span> <span m="2564760">will</span> <span m="2564910">have</span> <span m="2566286">the</span> <span m="2566740">8</span> <span m="2567050">of</span> <span m="2567340">spades</span> <span m="2568350">and</span> <span m="2568530">then</span> <span m="2568790">the</span> <span m="2568880">9</span> <span m="2569310">of</span> <span m="2569450">diamonds</span> <span m="2570600">and</span> <span m="2570800">all</span> <span m="2570930">the</span> <span m="2571060">other</span> <span m="2571260">8's--</span> <span m="2571890">the 8 of</span> <span m="2572030">diamonds</span> <span m="2572930">and the</span> <span m="2573080">8</span> <span m="2573370">of</span> <span m="2573450">hearts</span> <span m="2574610">and,</span> <span m="2574780">say,</span> <span m="2575035">the 8</span> <span m="2575290">of</span> <span m="2575470">clubs.</span> <span m="2578440">And</span> <span m="2579310">how</span> <span m="2579540">do</span> <span m="2579680">we</span> <span m="2579830">count</span> <span m="2580290">these</span> <span m="2580550">types</span> <span m="2580870">of</span> <span m="2580970">hands?</span> <span m="2581820">We're going</span> <span m="2582100">to</span> <span m="2582230">look</span> <span m="2582380">for</span> <span m="2582530">a</span> <span m="2582620">representation</span> <span m="2584620">of</span> <span m="2584780">how</span> <span m="2585090">can</span> <span m="2585260">you</span> <span m="2585430">represent</span> <span m="2586100">such</span> <span m="2586520">hands,</span> <span m="2586790">such</span> <span m="2587620">objects?</span> <span m="2588070">So</span> <span m="2588180">we</span> <span m="2588300">have</span> <span m="2588460">to</span> <span m="2588590">count</span> <span m="2588870">a</span> <span m="2588920">special</span> <span m="2589340">type</span> <span m="2589600">of</span> <span m="2589690">object,</span> <span m="2591120">a</span> <span m="2591220">special</span> <span m="2591690">kind</span> <span m="2591940">of</span> <span m="2592050">poker</span> <span m="2592420">hand.</span> <span m="2593310">And</span> <span m="2594490">in</span> <span m="2594690">order</span> <span m="2594830">to</span> <span m="2594970">do</span> <span m="2595120">that,</span> <span m="2595360">we're</span> <span m="2595590">going</span> <span m="2595800">to</span> <span m="2595990">look</span> <span m="2596170">for</span> <span m="2596680">a</span> <span m="2597250">way</span> <span m="2597510">to</span> <span m="2597740">represent</span> <span m="2598570">these</span> <span m="2598920">objects</span> <span m="2600190">and</span> <span m="2600330">in</span> <span m="2600510">such</span> <span m="2600860">a</span> <span m="2600950">way</span> <span m="2601100">that</span> <span m="2601260">we</span> <span m="2601380">can</span> <span m="2601600">count</span> <span m="2601860">them</span> <span m="2601990">very</span> <span m="2602340">easily.</span> </p>
<p><span m="2604450">And</span> <span m="2604600">that's</span> <span m="2605030">really</span> <span m="2605340">the</span> <span m="2605520">trick</span> <span m="2606160">in</span> <span m="2606330">order</span> <span m="2606530">to</span> <span m="2606680">solve</span> <span m="2607030">these</span> <span m="2607170">kinds</span> <span m="2607420">of</span> <span m="2607510">counting</span> <span m="2607870">problems.</span> <span m="2609500">So the</span> <span m="2609830">representation</span> <span m="2610660">that</span> <span m="2610770">we</span> <span m="2610900">have</span> <span m="2611140">here</span> <span m="2611910">is--</span> <span m="2612680">well,</span> <span m="2612860">we</span> <span m="2612910">can</span> <span m="2613190">choose--</span> <span m="2615820">first</span> <span m="2616160">of</span> <span m="2616210">all,</span> <span m="2616420">we</span> <span m="2616530">can</span> <span m="2616710">choose</span> <span m="2617070">the</span> <span m="2617280">value</span> <span m="2619840">of</span> <span m="2621070">the</span> <span m="2621270">four</span> <span m="2621560">kinds--</span> <span m="2622910">of</span> <span m="2622990">the</span> <span m="2623270">four</span> <span m="2623530">cards.</span> <span m="2626430">So</span> <span m="2626650">how</span> <span m="2626820">many</span> <span m="2627080">choices</span> <span m="2627530">do</span> <span m="2627720">I</span> <span m="2627820">have?</span> <span m="2629970">Well,</span> <span m="2630260">I</span> <span m="2630320">got</span> <span m="2632040">any</span> <span m="2632310">choice.</span> <span m="2633820">There</span> <span m="2633980">are</span> <span m="2634030">13</span> <span m="2634570">different</span> <span m="2634920">values</span> <span m="2636140">so</span> <span m="2636480">I</span> <span m="2636570">have</span> <span m="2636810">13</span> <span m="2637240">choices.</span> <span m="2637990">So</span> <span m="2638270">that's</span> <span m="2638510">easy.</span> </p>
<p><span m="2641260">Secondly,</span> <span m="2643280">I</span> <span m="2643400">can</span> <span m="2643620">choose</span> <span m="2643910">the</span> <span m="2644020">value</span> <span m="2644600">of</span> <span m="2645440">the</span> <span m="2645660">extra</span> <span m="2646140">card.</span> <span m="2650920">So</span> <span m="2651100">how</span> <span m="2651230">many</span> <span m="2651440">choices</span> <span m="2651790">do</span> <span m="2651910">I</span> <span m="2652020">have</span> <span m="2652220">here?</span> <span m="2653310">Well,</span> <span m="2655540">the</span> <span m="2655660">four</span> <span m="2656040">of a</span> <span m="2656190">kind</span> <span m="2656610">already</span> <span m="2656930">eliminates</span> <span m="2657500">one</span> <span m="2657740">kind</span> <span m="2658000">of</span> <span m="2658160">value</span> <span m="2658450">completely.</span> <span m="2659100">Right?</span> <span m="2659390">So</span> <span m="2659590">I</span> <span m="2659690">have</span> <span m="2661100">any</span> <span m="2661470">of</span> <span m="2661790">the</span> <span m="2661920">other</span> <span m="2662120">values</span> <span m="2663730">is</span> <span m="2663870">possible.</span> <span m="2665020">Is</span> <span m="2665220">equally</span> <span m="2665600">likely,</span> <span m="2665940">even.</span> <span m="2667370">So</span> <span m="2667410">I have</span> <span m="2667740">12</span> <span m="2669140">choices.</span> </p>
<p><span m="2670670">And,</span> <span m="2672460">lastly,</span> <span m="2673050">I</span> <span m="2673130">can</span> <span m="2673350">represent--</span> <span m="2674240">I</span> <span m="2674330">need</span> <span m="2674510">to</span> <span m="2674620">still</span> <span m="2674840">represent</span> <span m="2675870">the</span> <span m="2676120">suit</span> <span m="2676320">of the</span> <span m="2676430">extra</span> <span m="2676890">card.</span> <span m="2677280">So</span> <span m="2678140">I</span> <span m="2678280">also</span> <span m="2678540">want</span> <span m="2678720">to</span> <span m="2678850">know</span> <span m="2679840">the</span> <span m="2679930">suit</span> <span m="2681240">of</span> <span m="2681470">the</span> <span m="2681600">extra</span> <span m="2681980">card.</span> <span m="2683300">And</span> <span m="2683440">how</span> <span m="2683560">many</span> <span m="2683770">choices</span> <span m="2684100">do</span> <span m="2684220">I have?</span> <span m="2684570">Well, I</span> <span m="2684670">can</span> <span m="2684860">choose</span> <span m="2685140">any</span> <span m="2685460">of</span> <span m="2685620">those</span> <span m="2685850">four.</span> <span m="2687080">So</span> <span m="2687290">that's</span> <span m="2687540">four.</span> </p>
<p><span m="2689970">So,</span> <span m="2690170">essentially,</span> <span m="2690620">what</span> <span m="2690800">we</span> <span m="2690970">constructed</span> <span m="2691590">here</span> <span m="2692110">is</span> <span m="2692550">a</span> <span m="2692990">mapping</span> <span m="2694440">by</span> <span m="2694700">using</span> <span m="2695040">this</span> <span m="2695210">representation.</span> <span m="2697830">And</span> <span m="2697980">the</span> <span m="2698050">mapping</span> <span m="2698690">tells</span> <span m="2699180">us--</span> <span m="2702900">is</span> <span m="2703070">as</span> <span m="2703230">follows.</span> <span m="2706040">Let's</span> <span m="2706280">keep</span> <span m="2706540">this</span> <span m="2706820">up.</span> <span m="2715330">So</span> <span m="2715600">the</span> <span m="2715690">mapping</span> <span m="2716770">goes</span> <span m="2717140">from</span> <span m="2718040">poker</span> <span m="2718510">hands--</span> <span m="2719160">from</span> <span m="2719420">card--</span> <span m="2719910">from</span> <span m="2720900">hands</span> <span m="2721390">to--</span> <span m="2724110">from</span> <span m="2724330">hands</span> <span m="2724640">with</span> <span m="2724830">four of a</span> <span m="2725210">kind</span> <span m="2726050">to</span> <span m="2726160">this</span> <span m="2726400">representation.</span> <span m="2727940">So</span> <span m="2728150">let's</span> <span m="2728700">write</span> <span m="2729000">it</span> <span m="2729140">out.</span> </p>
<p><span m="2729480">So</span> <span m="2729490">we</span> <span m="2729590">have</span> <span m="2730530">four</span> <span m="2730910">of a</span> <span m="2731200">kind.</span> <span m="2733960">And</span> <span m="2734120">then</span> <span m="2734240">we</span> <span m="2734340">have</span> <span m="2734600">a</span> <span m="2734650">function--</span> <span m="2735510">a</span> <span m="2735680">mapping</span> <span m="2736080">f--</span> <span m="2736940">that</span> <span m="2737150">goes</span> <span m="2737450">to</span> <span m="2737540">this</span> <span m="2737750">representation.</span> <span m="2739240">We</span> <span m="2739410">will</span> <span m="2739620">have</span> <span m="2739910">the</span> <span m="2740010">first</span> <span m="2740450">entry--</span> <span m="2741010">the</span> <span m="2741310">value</span> <span m="2741480">1</span> <span m="2743020">and</span> <span m="2743200">the</span> <span m="2743270">value</span> <span m="2743690">2</span> <span m="2744760">and</span> <span m="2744930">then</span> <span m="2745060">a</span> <span m="2745140">value</span> <span m="2745530">3.</span> <span m="2748930">So</span> <span m="2749130">for</span> <span m="2749310">example,</span> <span m="2750380">if</span> <span m="2750510">you</span> <span m="2750660">take</span> <span m="2751060">this</span> <span m="2753070">hands</span> <span m="2753450">of</span> <span m="2753630">cards,</span> <span m="2754780">well,</span> <span m="2755290">the</span> <span m="2755410">first</span> <span m="2755890">one,</span> <span m="2757750">we</span> <span m="2757760">see</span> <span m="2758000">that</span> <span m="2758160">the</span> <span m="2758240">four of a</span> <span m="2758700">kind</span> <span m="2759080">has</span> <span m="2759330">value</span> <span m="2759800">8.</span> <span m="2761820">The</span> <span m="2761960">value</span> <span m="2762260">of</span> <span m="2762390">the</span> <span m="2762450">second</span> <span m="2762850">card</span> <span m="2763120">is</span> <span m="2763450">the</span> <span m="2764110">9</span> <span m="2764620">of the</span> <span m="2764950">9</span> <span m="2765180">diamonds</span> <span m="2765630">cards.</span> <span m="2766860">So we have</span> <span m="2767260">9.</span> <span m="2768520">And</span> <span m="2769310">finally,</span> <span m="2769730">the</span> <span m="2769860">suit</span> <span m="2770590">that</span> <span m="2770720">I</span> <span m="2770800">need</span> <span m="2770970">to</span> <span m="2771050">select</span> <span m="2771400">for, the</span> <span m="2771760">extra</span> <span m="2771970">card</span> <span m="2772990">is</span> <span m="2773330">diamonds.</span> <span m="2773910">So</span> <span m="2774050">this</span> <span m="2774320">is an</span> <span m="2774590">example.</span> </p>
<p><span m="2778120">And</span> <span m="2779780">now</span> <span m="2780010">we</span> <span m="2780130">know,</span> <span m="2780360">because</span> <span m="2780630">this</span> <span m="2780800">is</span> <span m="2780890">a</span> <span m="2780960">bijection,</span> <span m="2781940">we</span> <span m="2782060">know</span> <span m="2782270">that</span> <span m="2782450">the</span> <span m="2782530">number</span> <span m="2782840">of</span> <span m="2783520">hands</span> <span m="2783920">with</span> <span m="2784180">four of a</span> <span m="2784620">kind</span> <span m="2785030">is</span> <span m="2785440">equal</span> <span m="2786680">to</span> <span m="2787660">all</span> <span m="2787920">these</span> <span m="2788140">types</span> <span m="2788440">of</span> <span m="2788510">sequences,</span> <span m="2789340">all</span> <span m="2789500">these</span> <span m="2789700">types</span> <span m="2790050">of</span> <span m="2790150">sequences</span> <span m="2791290">that</span> <span m="2791460">can--</span> <span m="2793780">that</span> <span m="2793870">are</span> <span m="2793980">chosen</span> <span m="2794420">according</span> <span m="2794780">to</span> <span m="2794940">this</span> <span m="2795210">representation.</span> <span m="2797860">Well,</span> <span m="2798040">we</span> <span m="2798190">have</span> <span m="2798630">13</span> <span m="2799260">choices</span> <span m="2799750">for</span> <span m="2799890">the</span> <span m="2799980">first</span> <span m="2800730">value.</span> <span m="2801970">And</span> <span m="2802200">given</span> <span m="2802510">this</span> <span m="2802720">first</span> <span m="2803050">value,</span> <span m="2804080">we</span> <span m="2804270">have</span> <span m="2804930">12</span> <span m="2805320">choices</span> <span m="2806060">for</span> <span m="2806250">the</span> <span m="2806350">second</span> <span m="2806910">value.</span> <span m="2807750">And</span> <span m="2808090">given</span> <span m="2808400">those</span> <span m="2808720">two,</span> <span m="2808950">I</span> <span m="2809050">have</span> <span m="2809230">four</span> <span m="2809550">choices</span> <span m="2810280">for</span> <span m="2810730">the</span> <span m="2810820">very</span> <span m="2811060">last</span> <span m="2813430">entry</span> <span m="2813840">in</span> <span m="2813930">this</span> <span m="2814330">sequence.</span> </p>
<p><span m="2815870">So</span> <span m="2816170">this</span> <span m="2816360">is</span> <span m="2816540">the</span> <span m="2816650">generalized</span> <span m="2817160">product</span> <span m="2817460">rule</span> <span m="2817720">that</span> <span m="2817850">I'm</span> <span m="2817930">using.</span> <span m="2819100">So</span> <span m="2819340">we</span> <span m="2819440">see</span> <span m="2819700">that</span> <span m="2819870">the</span> <span m="2819970">number</span> <span m="2820250">of</span> <span m="2820370">sequences</span> <span m="2822830">is</span> <span m="2823160">equal</span> <span m="2824210">to</span> <span m="2824860">13</span> <span m="2825560">times</span> <span m="2825950">12</span> <span m="2826480">times</span> <span m="2826840">4.</span> <span m="2828170">Turns</span> <span m="2828490">out</span> <span m="2828640">this</span> <span m="2828800">is</span> <span m="2828900">equal</span> <span m="2829130">to</span> <span m="2829250">624.</span> <span m="2830650">Well,</span> <span m="2830990">if</span> <span m="2831100">you</span> <span m="2831170">divide</span> <span m="2831445">it</span> <span m="2831720">over</span> <span m="2832100">total</span> <span m="2832410">number</span> <span m="2832650">of</span> <span m="2832810">cards.</span> </p>
<p><span m="2835250">So</span> <span m="2835520">how</span> <span m="2835670">did</span> <span m="2835880">we do</span> <span m="2836030">this?</span> <span m="2836630">This is</span> <span m="2836820">the</span> <span m="2836950">generalized</span> <span m="2837210">product</span> <span m="2837520">rule.</span> <span m="2841500">If</span> <span m="2841680">you</span> <span m="2841750">divide</span> <span m="2842590">this</span> <span m="2842780">number</span> <span m="2843000">of</span> <span m="2843170">the</span> <span m="2843230">total</span> <span m="2843540">number</span> <span m="2843860">of</span> <span m="2844230">cards</span> <span m="2844560">we could</span> <span m="2844890">get</span> <span m="2845050">the</span> <span m="2845130">fraction</span> <span m="2845540">of</span> <span m="2845710">one</span> <span m="2846010">over</span> <span m="2846380">about</span> <span m="2846730">4,000.</span> <span m="2848030">So</span> <span m="2848230">it's</span> <span m="2848360">really</span> <span m="2848680">rare</span> <span m="2848970">that</span> <span m="2849060">you</span> <span m="2849150">get</span> <span m="2849320">four of a</span> <span m="2849730">kind.</span> <span m="2850045">So</span> <span m="2850360">it's</span> <span m="2850520">a</span> <span m="2850580">really</span> <span m="2850790">good</span> <span m="2850980">hand.</span> </p>
<p><span m="2853550">OK.</span> <span m="2853820">Let's</span> <span m="2854340">do</span> <span m="2854540">a</span> <span m="2854590">few</span> <span m="2854840">more</span> <span m="2855620">of</span> <span m="2855950">these.</span> <span m="2864180">We</span> <span m="2864300">also</span> <span m="2864490">like</span> <span m="2864680">to</span> <span m="2864800">know</span> <span m="2865110">how</span> <span m="2865400">many</span> <span m="2865900">full</span> <span m="2866260">houses</span> <span m="2866690">there</span> <span m="2866900">are.</span> <span m="2867520">A</span> <span m="2867600">full</span> <span m="2867830">house</span> <span m="2868140">is</span> <span m="2868300">a</span> <span m="2868340">special</span> <span m="2868810">hand.</span> <span m="2869270">It</span> <span m="2869620">has</span> <span m="2869880">three</span> <span m="2871690">cards</span> <span m="2871950">of</span> <span m="2872170">one</span> <span m="2872820">value</span> <span m="2873370">and</span> <span m="2873500">two</span> <span m="2873710">cards</span> <span m="2874000">of</span> <span m="2874110">another</span> <span m="2874370">value.</span> </p>
<p><span m="2890280">So</span> <span m="2890560">how</span> <span m="2890680">many</span> <span m="2890970">are</span> <span m="2891070">there?</span> <span m="2892290">So</span> <span m="2892520">again,</span> <span m="2892790">we</span> <span m="2892900">are</span> <span m="2893140">going to</span> <span m="2893290">use</span> <span m="2893500">the exact</span> <span m="2893900">same</span> <span m="2894110">principle.</span> <span m="2895115">We're</span> <span m="2895490">going</span> <span m="2895780">to</span> <span m="2896500">find</span> <span m="2896850">a</span> <span m="2896890">representation</span> <span m="2897850">of</span> <span m="2898010">this</span> <span m="2898250">type</span> <span m="2898520">of</span> <span m="2898640">hand.</span> <span m="2900020">So</span> <span m="2900220">we</span> <span m="2900310">have</span> <span m="2900430">three</span> <span m="2900680">cards</span> <span m="2902260">of</span> <span m="2903010">one</span> <span m="2903350">value</span> <span m="2906550">and</span> <span m="2906860">two</span> <span m="2907110">cards</span> <span m="2909750">of</span> <span m="2910160">another.</span> </p>
<p><span m="2916690">So</span> <span m="2916870">for</span> <span m="2917030">example,</span> <span m="2919850">we</span> <span m="2919910">may</span> <span m="2920190">have</span> <span m="2921670">the</span> <span m="2921810">hands</span> <span m="2922290">that</span> <span m="2922420">contain,</span> <span m="2922990">say,</span> <span m="2923740">three</span> <span m="2924720">cards</span> <span m="2924990">with</span> <span m="2925250">value</span> <span m="2925570">2.</span> <span m="2926060">A</span> <span m="2926550">2 clubs,</span> <span m="2927100">a</span> <span m="2927220">2</span> <span m="2927410">spades,</span> <span m="2928710">and</span> <span m="2928960">also</span> <span m="2929440">a 2</span> <span m="2929620">diamonds.</span> <span m="2930970">And,</span> <span m="2931210">say,</span> <span m="2931470">a Jack</span> <span m="2931880">club</span> <span m="2932450">and</span> <span m="2932830">a</span> <span m="2932890">Jack</span> <span m="2933920">of</span> <span m="2933960">diamonds.</span> <span m="2935520">And</span> <span m="2937240">another</span> <span m="2937550">example</span> <span m="2938020">could</span> <span m="2938220">be,</span> <span m="2938610">say,</span> <span m="2939130">a 5</span> <span m="2939490">of</span> <span m="2939850">diamonds</span> <span m="2940470">and</span> <span m="2940710">a</span> <span m="2940820">5</span> <span m="2941330">of</span> <span m="2941410">hearts,</span> <span m="2942880">5</span> <span m="2943430">of</span> <span m="2943600">clubs,</span> <span m="2944270">7</span> <span m="2944700">of</span> <span m="2944820">hearts,</span> <span m="2945860">and</span> <span m="2946250">also</span> <span m="2946680">a</span> <span m="2946780">7</span> <span m="2947080">of</span> <span m="2947960">club.</span> </p>
<p><span m="2950000">So</span> <span m="2950150">now</span> <span m="2950290">you</span> <span m="2950410">can</span> <span m="2950560">see</span> <span m="2950780">that</span> <span m="2951040">it's very easy</span> <span m="2951510">to</span> <span m="2951640">represent</span> <span m="2952180">this.</span> <span m="2952390">Right?</span> <span m="2955700">We</span> <span m="2955820">can</span> <span m="2955980">start</span> <span m="2956870">grouping</span> <span m="2957250">things</span> <span m="2957530">together.</span> <span m="2958020">We</span> <span m="2958120">can</span> <span m="2958410">say,</span> <span m="2959020">represent</span> <span m="2959560">this</span> <span m="2959750">by</span> <span m="2959890">first</span> <span m="2960330">taking</span> <span m="2961450">the</span> <span m="2961740">value</span> <span m="2962630">of</span> <span m="2963098">the</span> <span m="2966010">three</span> <span m="2966320">cards</span> <span m="2966880">that</span> <span m="2967050">I</span> <span m="2967120">have</span> <span m="2967410">here.</span> <span m="2967810">So</span> <span m="2967910">it's</span> <span m="2968120">2.</span> <span m="2969630">Then</span> <span m="2969850">I</span> <span m="2969890">want</span> <span m="2970110">to</span> <span m="2970330">indicate</span> <span m="2971020">in</span> <span m="2971100">a</span> <span m="2971160">second</span> <span m="2971980">term</span> <span m="2972270">in</span> <span m="2972360">my</span> <span m="2972500">sequence--</span> <span m="2973330">in</span> <span m="2973460">my</span> <span m="2973560">representation--</span> <span m="2974850">which</span> <span m="2975260">suits</span> <span m="2975830">did</span> <span m="2976070">I</span> <span m="2976240">use</span> <span m="2977390">over</span> <span m="2977710">here?</span> <span m="2977910">Well,</span> <span m="2978160">I used</span> <span m="2979020">clubs,</span> <span m="2979350">spades</span> <span m="2979890">and</span> <span m="2980020">diamonds.</span> <span m="2980490">So</span> <span m="2980670">it's</span> <span m="2980870">a</span> <span m="2980920">set</span> <span m="2982430">club,</span> <span m="2983250">spades</span> <span m="2984390">and</span> <span m="2984720">diamonds.</span> </p>
<p><span m="2986560">And</span> <span m="2987330">now</span> <span m="2988150">I also</span> <span m="2988290">have a</span> <span m="2988430">pair.</span> <span m="2989180">Like,</span> <span m="2990190">two</span> <span m="2990430">cards</span> <span m="2990760">with</span> <span m="2990820">the</span> <span m="2990890">same</span> <span m="2991160">value.</span> <span m="2991550">What's</span> <span m="2991780">the</span> <span m="2991870">value?</span> <span m="2992960">It's</span> <span m="2993270">Jack.</span> <span m="2993990">And</span> <span m="2995350">what</span> <span m="2995640">are</span> <span m="2995740">the suits</span> <span m="2996160">associated</span> <span m="2996750">to</span> <span m="2997010">these?</span> <span m="2997630">Well,</span> <span m="2998120">clubs</span> <span m="2998640">and</span> <span m="2998920">the</span> <span m="2999000">diamonds.</span> <span m="3000870">So</span> <span m="3002730">I have</span> <span m="3002870">not</span> <span m="3003010">yet</span> <span m="3003270">for--</span> <span m="3003880">given</span> <span m="3004060">you</span> <span m="3004230">the</span> <span m="3004380">formal</span> <span m="3004700">definition</span> <span m="3005200">of</span> <span m="3005330">the</span> <span m="3005370">representation,</span> <span m="3006060">but</span> <span m="3006170">you</span> <span m="3006230">can</span> <span m="3006380">see</span> <span m="3006940">that</span> <span m="3007060">you</span> <span m="3007160">could</span> <span m="3007320">do</span> <span m="3007450">something</span> <span m="3007820">like</span> <span m="3008040">this.</span> <span m="3008440">So</span> <span m="3008460">this is</span> <span m="3008860">often</span> <span m="3009290">how</span> <span m="3009450">we</span> <span m="3009530">start</span> <span m="3009860">out</span> <span m="3010170">on</span> <span m="3010240">a</span> <span m="3010290">piece</span> <span m="3010550">of</span> <span m="3010680">paper.</span> <span m="3011100">You</span> <span m="3011230">try</span> <span m="3011450">to</span> <span m="3012050">do</span> <span m="3012210">something</span> <span m="3013410">and</span> <span m="3014720">hopefully</span> <span m="3015240">it</span> <span m="3015370">works.</span> </p>
<p><span m="3017390">So</span> <span m="3017590">we</span> <span m="3017720">have--</span> <span m="3018290">here</span> <span m="3018550">we</span> <span m="3018660">have</span> <span m="3019720">diamonds,</span> <span m="3020200">hearts,</span> <span m="3020460">and</span> <span m="3020720">clubs.</span> <span m="3023850">And</span> <span m="3024210">then</span> <span m="3024300">we</span> <span m="3024490">have--</span> <span m="3024780">finally</span> <span m="3025470">we</span> <span m="3025590">have</span> <span m="3027680">two</span> <span m="3027860">cards</span> <span m="3028160">with</span> <span m="3028220">the</span> <span m="3028290">same</span> <span m="3028580">value</span> <span m="3028890">7.</span> <span m="3030150">And</span> <span m="3030390">they</span> <span m="3030600">have</span> <span m="3030930">the</span> <span m="3031000">hearts</span> <span m="3031275">and</span> <span m="3031550">clubs</span> <span m="3032210">as</span> <span m="3033880">suits.</span> <span m="3037540">OK.</span> </p>
<p><span m="3039170">So</span> <span m="3039440">the</span> <span m="3039700">representation</span> <span m="3042010">is</span> <span m="3046710">defined</span> <span m="3047150">as</span> <span m="3047300">follows.</span> <span m="3048310">It</span> <span m="3048450">just</span> <span m="3049220">started</span> <span m="3049750">out</span> <span m="3054580">with</span> <span m="3055180">a</span> <span m="3055320">first</span> <span m="3056590">entry</span> <span m="3057280">in</span> <span m="3057460">my</span> <span m="3057710">representation.</span> <span m="3059830">And</span> <span m="3059990">this</span> <span m="3060170">is</span> <span m="3060400">going</span> <span m="3060610">to</span> <span m="3060710">be</span> <span m="3060850">the</span> <span m="3061040">value</span> <span m="3062600">of</span> <span m="3062900">the</span> <span m="3063010">triple.</span> <span m="3066670">So</span> <span m="3066790">how</span> <span m="3066950">many</span> <span m="3067200">choices</span> <span m="3067680">do we</span> <span m="3067850">have</span> <span m="3068130">for</span> <span m="3068260">this</span> <span m="3068490">value?</span> <span m="3068980">Well, I can</span> <span m="3069140">choose</span> <span m="3069380">any</span> <span m="3069720">of</span> <span m="3069860">the</span> <span m="3069940">13</span> <span m="3070370">possible</span> <span m="3070800">values.</span> <span m="3071330">So I have</span> <span m="3071610">13</span> <span m="3072060">choices.</span> </p>
<p><span m="3074890">And</span> <span m="3076860">the</span> <span m="3076970">second</span> <span m="3077290">part</span> <span m="3077720">is</span> <span m="3078100">the</span> <span m="3078220">suits</span> <span m="3080150">of</span> <span m="3080720">the</span> <span m="3081060">triple.</span> <span m="3083220">Well,</span> <span m="3084440">I</span> <span m="3084650">have</span> <span m="3084980">to</span> <span m="3085170">choose</span> <span m="3085660">a</span> <span m="3086400">subset</span> <span m="3087300">of</span> <span m="3087730">the</span> <span m="3087830">four</span> <span m="3088450">possible</span> <span m="3088870">suits</span> <span m="3089350">of</span> <span m="3089450">size</span> <span m="3090070">three.</span> <span m="3090300">So</span> <span m="3090410">I</span> <span m="3090490">need</span> <span m="3090660">to</span> <span m="3090790">have</span> <span m="3090950">a</span> <span m="3091000">subset</span> <span m="3091410">of</span> <span m="3091480">size</span> <span m="3091930">three</span> <span m="3093080">out</span> <span m="3093310">of</span> <span m="3093510">the</span> <span m="3093600">four</span> <span m="3093870">possible</span> <span m="3094320">suits.</span> <span m="3095690">And</span> <span m="3095830">that</span> <span m="3096030">will</span> <span m="3097050">give</span> <span m="3097320">me</span> <span m="3097590">the</span> <span m="3097700">proper</span> <span m="3097960">representation.</span> </p>
<p><span m="3098830">So</span> <span m="3099610">how</span> <span m="3099780">many</span> <span m="3100080">choices</span> <span m="3100550">do</span> <span m="3100680">I</span> <span m="3100770">have</span> <span m="3101000">for</span> <span m="3101110">such</span> <span m="3101350">a</span> <span m="3101410">subset?</span> <span m="3102320">Well,</span> <span m="3102490">there's</span> <span m="3102700">three</span> <span m="3102910">elements</span> <span m="3103450">out</span> <span m="3103680">of</span> <span m="3103780">four.</span> <span m="3106190">So</span> <span m="3106440">we</span> <span m="3106560">can</span> <span m="3106740">use</span> <span m="3107060">the</span> <span m="3107340">subset</span> <span m="3107730">rule</span> <span m="3108620">and</span> <span m="3108810">see</span> <span m="3109590">that</span> <span m="3109790">this</span> <span m="3110170">has</span> <span m="3110840">4</span> <span m="3111510">choose</span> <span m="3111880">3</span> <span m="3112490">equals</span> <span m="3113340">four</span> <span m="3115310">choices.</span> <span m="3118370">Then</span> <span m="3119890">the</span> <span m="3120010">last</span> <span m="3120370">part</span> <span m="3120730">is</span> <span m="3120950">to</span> <span m="3121050">value</span> <span m="3121620">of</span> <span m="3121910">the</span> <span m="3122100">pair</span> <span m="3122560">that</span> <span m="3122740">I</span> <span m="3122850">have</span> <span m="3123480">of</span> <span m="3123640">the</span> <span m="3123800">two</span> <span m="3124060">cards</span> <span m="3124530">that are</span> <span m="3125140">having</span> <span m="3125410">the</span> <span m="3125490">same</span> <span m="3125760">value.</span> </p>
<p><span m="3129240">Now,</span> <span m="3129370">how</span> <span m="3129530">many</span> <span m="3129750">choices</span> <span m="3130090">do</span> <span m="3130260">I</span> <span m="3130380">have</span> <span m="3130650">here?</span> <span m="3131920">Well,</span> <span m="3132880">I have to</span> <span m="3132970">be</span> <span m="3133130">a</span> <span m="3133180">little</span> <span m="3133350">bit</span> <span m="3133510">careful,</span> <span m="3133785">though.</span> <span m="3134160">I</span> <span m="3134260">just</span> <span m="3134440">want</span> <span m="3134570">to</span> <span m="3134660">make</span> <span m="3134840">sure</span> <span m="3135180">that</span> <span m="3135320">I do</span> <span m="3135460">the</span> <span m="3135590">right</span> <span m="3135800">reasoning</span> <span m="3136290">here.</span> <span m="3138360">I</span> <span m="3138500">know</span> <span m="3138680">that</span> <span m="3138820">there's</span> <span m="3139010">still--</span> <span m="3140372">because</span> <span m="3140740">I</span> <span m="3140780">chose</span> <span m="3141150">a</span> <span m="3141320">triple</span> <span m="3142890">of</span> <span m="3143120">cards</span> <span m="3143400">of</span> <span m="3143510">the</span> <span m="3143610">same</span> <span m="3143890">value--</span> <span m="3145410">for</span> <span m="3145570">example,</span> <span m="3146310">over</span> <span m="3146630">here</span> <span m="3147590">I</span> <span m="3147730">chose</span> <span m="3148200">all</span> <span m="3148430">these</span> <span m="3148660">2's--</span> <span m="3148960">but</span> <span m="3149430">I</span> <span m="3149550">still</span> <span m="3149790">have</span> <span m="3149990">one</span> <span m="3150780">card</span> <span m="3151600">in</span> <span m="3151800">my</span> <span m="3151970">deck</span> <span m="3152510">which</span> <span m="3152750">has</span> <span m="3153060">a 2.</span> <span m="3154120">Actually,</span> <span m="3154560">here</span> <span m="3154770">I've</span> <span m="3154900">chosen</span> <span m="3155340">clubs,</span> <span m="3155620">spades</span> <span m="3155900">and</span> <span m="3156050">diamonds.</span> <span m="3156400">So</span> <span m="3156840">the</span> <span m="3156960">one</span> <span m="3157120">that is</span> <span m="3157410">missing</span> <span m="3157910">is</span> <span m="3158220">the</span> <span m="3158340">2</span> <span m="3158600">of</span> <span m="3158680">hearts.</span> <span m="3159910">So I</span> <span m="3159970">could</span> <span m="3160160">possibly</span> <span m="3160660">choose</span> <span m="3160970">the</span> <span m="3161050">2</span> <span m="3161260">of</span> <span m="3161360">hearts.</span> </p>
<p><span m="3162910">But</span> <span m="3163040">wait</span> <span m="3163240">a</span> <span m="3163270">minute.</span> <span m="3163710">If</span> <span m="3163890">I</span> <span m="3163950">choose</span> <span m="3164200">the</span> <span m="3164300">2</span> <span m="3164480">of</span> <span m="3164590">hearts,</span> <span m="3165950">how</span> <span m="3166120">can</span> <span m="3166300">I</span> <span m="3166330">make</span> <span m="3166580">a</span> <span m="3166710">pair?</span> <span m="3167350">That's</span> <span m="3167570">not</span> <span m="3167760">possible,</span> <span m="3168190">right?</span> <span m="3168430">Because</span> <span m="3168740">I've</span> <span m="3169480">already</span> <span m="3169910">chosen</span> <span m="3170300">all</span> <span m="3170400">the</span> <span m="3170520">other</span> <span m="3170730">2's</span> <span m="3171970">so</span> <span m="3172020">there's</span> <span m="3172220">no</span> <span m="3172580">other</span> <span m="3172920">2</span> <span m="3173330">to match</span> <span m="3174400">to</span> <span m="3174520">find</span> <span m="3174830">a</span> <span m="3174870">pair.</span> <span m="3175440">So I</span> <span m="3175680">actually</span> <span m="3176000">cannot</span> <span m="3176390">choose</span> <span m="3177740">this</span> <span m="3177960">particular</span> <span m="3178490">value</span> <span m="3178820">of</span> <span m="3178940">this</span> <span m="3179140">triple</span> <span m="3181360">for the</span> <span m="3181810">pair.</span> <span m="3183640">But</span> <span m="3183780">all</span> <span m="3183920">the</span> <span m="3184050">other</span> <span m="3184220">values</span> <span m="3184600">are</span> <span m="3184690">possible.</span> <span m="3185970">So</span> <span m="3186300">I</span> <span m="3186430">have</span> <span m="3187690">12</span> <span m="3188090">choices.</span> <span m="3189450">One</span> <span m="3189710">less</span> <span m="3190130">than</span> <span m="3190900">the</span> <span m="3191090">one</span> <span m="3191240">that I've</span> <span m="3191550">already</span> <span m="3191850">chosen</span> <span m="3192270">for</span> <span m="3192480">the</span> <span m="3192570">triple.</span> </p>
<p><span m="3195930">Now,</span> <span m="3196130">and</span> <span m="3196210">then</span> <span m="3196430">similarly,</span> <span m="3197520">I</span> <span m="3197670">can</span> <span m="3197880">choose</span> <span m="3198730">suits</span> <span m="3199410">of</span> <span m="3199665">the</span> <span m="3199920">pair.</span> <span m="3201370">And</span> <span m="3202580">how</span> <span m="3202700">many</span> <span m="3202910">choices</span> <span m="3203290">do</span> <span m="3203430">I</span> <span m="3203530">have?</span> <span m="3203810">Well,</span> <span m="3204000">look</span> <span m="3204210">at</span> <span m="3204640">every</span> <span m="3205840">possible</span> <span m="3206520">subset</span> <span m="3207230">that</span> <span m="3207380">I</span> <span m="3207440">can</span> <span m="3207670">have</span> <span m="3207930">of</span> <span m="3208070">size</span> <span m="3208570">two</span> <span m="3209260">out</span> <span m="3209470">of</span> <span m="3209610">the</span> <span m="3209700">four</span> <span m="3210070">suits.</span> <span m="3211230">So</span> <span m="3211540">I</span> <span m="3211630">use</span> <span m="3211860">the</span> <span m="3211910">subset</span> <span m="3212360">rule</span> <span m="3213500">and</span> <span m="3213710">find</span> <span m="3214240">that</span> <span m="3214410">this</span> <span m="3214640">is</span> <span m="3214830">4</span> <span m="3215160">choose</span> <span m="3215540">2</span> <span m="3217390">which</span> <span m="3217580">is</span> <span m="3217700">equal</span> <span m="3217960">to</span> <span m="3218120">6.</span> </p>
<p><span m="3220780">So</span> <span m="3220990">now</span> <span m="3221700">we're</span> <span m="3221860">going</span> <span m="3222060">to</span> <span m="3222170">multiply</span> <span m="3222840">all</span> <span m="3223110">of</span> <span m="3223220">these</span> <span m="3223430">together.</span> <span m="3224180">We</span> <span m="3224310">use</span> <span m="3224520">the</span> <span m="3224570">generalized</span> <span m="3225170">product</span> <span m="3225520">rule</span> <span m="3226060">again.</span> <span m="3227030">So</span> <span m="3228250">we</span> <span m="3229220">have</span> <span m="3229410">found</span> <span m="3229650">a</span> <span m="3229690">mapping</span> <span m="3230240">from</span> <span m="3230820">hands</span> <span m="3231250">with</span> <span m="3231420">a</span> <span m="3231480">full</span> <span m="3231760">house</span> <span m="3232640">to</span> <span m="3233060">these</span> <span m="3233520">types</span> <span m="3233830">of</span> <span m="3233950">representations--</span> <span m="3235070">to</span> <span m="3235610">this</span> <span m="3235800">representation.</span> <span m="3237040">And</span> <span m="3237670">this</span> <span m="3237920">mapping</span> <span m="3238220">is</span> <span m="3238360">bijective.</span> <span m="3239450">So</span> <span m="3239710">the</span> <span m="3240290">number</span> <span m="3240670">of</span> <span m="3240800">full</span> <span m="3241050">house</span> <span m="3241370">is</span> <span m="3241520">exactly</span> <span m="3242080">equal</span> <span m="3242270">to</span> <span m="3242370">the</span> <span m="3242470">number</span> <span m="3243210">of</span> <span m="3243480">these</span> <span m="3245020">representations.</span> <span m="3246940">And</span> <span m="3247360">by the</span> <span m="3247480">generalized</span> <span m="3247930">product</span> <span m="3248290">rule,</span> <span m="3248530">I</span> <span m="3248610">can</span> <span m="3248800">choose</span> <span m="3249090">the</span> <span m="3249160">first</span> <span m="3249560">entry</span> <span m="3250020">of</span> <span m="3250090">such</span> <span m="3250360">a</span> <span m="3250420">sequence</span> <span m="3251240">in</span> <span m="3251430">13</span> <span m="3251920">ways.</span> <span m="3252210">The</span> <span m="3252350">second</span> <span m="3252730">one,</span> <span m="3253040">given</span> <span m="3253320">the</span> <span m="3253410">first</span> <span m="3253780">one,</span> <span m="3253940">in</span> <span m="3254030">four</span> <span m="3254350">ways.</span> <span m="3255960">This</span> <span m="3256180">one</span> <span m="3256340">I</span> <span m="3256370">can</span> <span m="3256580">choose</span> <span m="3257000">in</span> <span m="3257170">12</span> <span m="3257580">ways</span> <span m="3257980">given</span> <span m="3259160">I've</span> <span m="3259400">already</span> <span m="3259500">chosen</span> <span m="3259870">my</span> <span m="3259980">triple.</span> <span m="3260760">And</span> <span m="3260980">so</span> <span m="3261230">on.</span> </p>
<p><span m="3261730">So</span> <span m="3261900">by</span> <span m="3262070">the</span> <span m="3262160">generalized</span> <span m="3262610">product</span> <span m="3262970">rule,</span> <span m="3263580">I</span> <span m="3263720">know</span> <span m="3264020">now</span> <span m="3264750">that</span> <span m="3264940">the</span> <span m="3265030">product</span> <span m="3265450">of</span> <span m="3265570">those</span> <span m="3265850">four</span> <span m="3266290">is</span> <span m="3266510">equal</span> <span m="3266850">to</span> <span m="3266960">the</span> <span m="3267120">total</span> <span m="3267500">number</span> <span m="3267810">of</span> <span m="3267900">full</span> <span m="3268190">house</span> <span m="3268570">hands.</span> <span m="3270740">So</span> <span m="3271010">how</span> <span m="3271130">much</span> <span m="3271390">is</span> <span m="3271580">that?</span> <span m="3271940">It's</span> <span m="3273310">13</span> <span m="3274200">times</span> <span m="3274560">4</span> <span m="3274820">choose</span> <span m="3275200">3</span> <span m="3276050">times</span> <span m="3276410">12</span> <span m="3276970">times</span> <span m="3277310">4</span> <span m="3277650">choose</span> <span m="3278500">2.</span> <span m="3279710">And</span> <span m="3279890">this</span> <span m="3280060">turns</span> <span m="3280400">out</span> <span m="3280560">to</span> <span m="3280660">be</span> <span m="3280910">equal</span> <span m="3281300">to</span> <span m="3282210">3,744.</span> <span m="3284910">Which</span> <span m="3285110">is</span> <span m="3285230">a</span> <span m="3285280">factor</span> <span m="3285650">six</span> <span m="3286060">bigger</span> <span m="3288360">than</span> <span m="3288590">a</span> <span m="3288890">four of</span> <span m="3288920">a kind.</span> <span m="3289410">So</span> <span m="3289570">it's</span> <span m="3289710">much</span> <span m="3289930">more</span> <span m="3290080">likely</span> <span m="3290490">that you</span> <span m="3290720">get</span> <span m="3290920">one</span> <span m="3291080">of</span> <span m="3291190">those.</span> <span m="3292060">And</span> <span m="3292230">that's</span> <span m="3292450">the</span> <span m="3292550">reason</span> <span m="3292910">why</span> <span m="3293030">four of</span> <span m="3293430">a</span> <span m="3293530">kind</span> <span m="3293790">has</span> <span m="3294050">more</span> <span m="3294290">worth--</span> <span m="3296430">is</span> <span m="3296900">worth</span> <span m="3297160">much</span> <span m="3297380">more</span> <span m="3298060">than</span> <span m="3298660">a</span> <span m="3298770">full</span> <span m="3299010">house.</span> </p>
<p><span m="3301410">So</span> <span m="3301650">let's</span> <span m="3301920">do</span> <span m="3302940">another</span> <span m="3303310">example,</span> <span m="3304220">a hand</span> <span m="3304810">with</span> <span m="3305590">two</span> <span m="3305840">pairs,</span> <span m="3306670">and</span> <span m="3306820">see</span> <span m="3306940">whether</span> <span m="3307090">we can</span> <span m="3307270">continue</span> <span m="3307830">this</span> <span m="3307990">type</span> <span m="3308220">of</span> <span m="3308310">reasoning.</span> <span m="3309190">It's</span> <span m="3309360">going</span> <span m="3309550">pretty</span> <span m="3309830">well.</span> <span m="3311010">And</span> <span m="3311150">maybe</span> <span m="3313960">we</span> <span m="3314070">can</span> <span m="3314430">do</span> <span m="3314540">the</span> <span m="3314660">same</span> <span m="3315143">thing.</span> <span m="3320940">We'll</span> <span m="3321190">see</span> <span m="3321480">that,</span> <span m="3323130">in</span> <span m="3323330">counting,</span> <span m="3323760">you</span> <span m="3323950">really</span> <span m="3324190">have</span> <span m="3324360">to</span> <span m="3324480">take</span> <span m="3324990">a</span> <span m="3325100">lot</span> <span m="3325370">of</span> <span m="3325540">care.</span> <span m="3326260">So</span> <span m="3326850">maybe</span> <span m="3327070">you</span> <span m="3327150">can</span> <span m="3327310">already</span> <span m="3327560">see</span> <span m="3328440">what's</span> <span m="3328630">happening</span> <span m="3329560">when</span> <span m="3329760">I</span> <span m="3329810">start</span> <span m="3330430">reasoning</span> <span m="3330890">in</span> <span m="3330980">the</span> <span m="3331100">exact</span> <span m="3331480">same</span> <span m="3331730">way</span> <span m="3332110">as</span> <span m="3332300">before.</span> </p>
<p><span m="3334130">So</span> <span m="3334400">let</span> <span m="3334550">me</span> <span m="3334640">first</span> <span m="3334910">define</span> <span m="3335290">what</span> <span m="3335450">I</span> <span m="3335490">want</span> <span m="3335690">to</span> <span m="3335820">count.</span> <span m="3336950">It's</span> <span m="3337130">a</span> <span m="3337860">hand</span> <span m="3339050">that</span> <span m="3339260">have</span> <span m="3339540">exactly</span> <span m="3340980">two</span> <span m="3341210">pairs.</span> <span m="3343750">So</span> <span m="3343880">what</span> <span m="3343970">does</span> <span m="3344150">that</span> <span m="3344320">mean?</span> <span m="3344850">It</span> <span m="3344930">means</span> <span m="3345350">that</span> <span m="3347450">there</span> <span m="3347600">is--</span> <span m="3348000">that</span> <span m="3348240">we</span> <span m="3348340">have</span> <span m="3348640">two</span> <span m="3348940">cards</span> <span m="3351630">of</span> <span m="3351850">one</span> <span m="3352170">value</span> <span m="3353680">and</span> <span m="3354030">another</span> <span m="3354350">two</span> <span m="3354570">cards</span> <span m="3355620">of</span> <span m="3356520">another</span> <span m="3356780">value.</span> </p>
<p><span m="3367900">So</span> <span m="3368120">let's</span> <span m="3369070">start</span> <span m="3369380">out</span> <span m="3369660">as</span> <span m="3369890">before.</span> <span m="3372190">We're going</span> <span m="3372440">to</span> <span m="3372540">write</span> <span m="3372730">out</span> <span m="3372900">a</span> <span m="3372990">representation</span> <span m="3374820">and</span> <span m="3374960">see</span> <span m="3377030">whether</span> <span m="3377140">we</span> <span m="3377250">can</span> <span m="3377430">do</span> <span m="3377630">this</span> <span m="3377850">properly.</span> <span m="3378660">So</span> <span m="3382180">we're going</span> <span m="3382540">to</span> <span m="3382720">use the</span> <span m="3383040">exact</span> <span m="3383830">same</span> <span m="3384130">technique.</span> <span m="3385470">So</span> <span m="3386840">first</span> <span m="3387130">of</span> <span m="3387190">all,</span> <span m="3387490">we're</span> <span m="3387600">going to</span> <span m="3387875">choose</span> <span m="3388150">the</span> <span m="3388240">value</span> <span m="3388940">of</span> <span m="3389570">the</span> <span m="3389740">first</span> <span m="3390090">pair.</span> </p>
<p><span m="3397080">Now,</span> <span m="3397250">how</span> <span m="3397390">many</span> <span m="3397670">ways</span> <span m="3397970">can</span> <span m="3398140">I</span> <span m="3398210">do</span> <span m="3398390">this?</span> <span m="3400580">Well,</span> <span m="3400750">I</span> <span m="3400800">can</span> <span m="3400970">do</span> <span m="3401070">this</span> <span m="3401260">in</span> <span m="3401370">13</span> <span m="3401950">ways.</span> <span m="3402770">Any</span> <span m="3403030">possible</span> <span m="3403420">value</span> <span m="3403910">is</span> <span m="3404090">possible.</span> <span m="3408320">I</span> <span m="3408480">need</span> <span m="3408700">to</span> <span m="3408810">choose</span> <span m="3409220">the</span> <span m="3409340">suits</span> <span m="3411000">of</span> <span m="3411770">the</span> <span m="3411910">first</span> <span m="3412350">pair.</span> </p>
<p><span m="3416280">And</span> <span m="3416470">how</span> <span m="3416580">many</span> <span m="3416830">ways</span> <span m="3417040">can</span> <span m="3417170">I</span> <span m="3417220">do</span> <span m="3417380">this?</span> <span m="3417740">Well,</span> <span m="3418010">I</span> <span m="3418150">use</span> <span m="3418360">the</span> <span m="3418440">same</span> <span m="3418700">techniques</span> <span m="3418955">as</span> <span m="3419210">over</span> <span m="3419470">here,</span> <span m="3419670">so</span> <span m="3419820">it's</span> <span m="3420210">any</span> <span m="3421370">possible</span> <span m="3421880">way</span> <span m="3422070">to</span> <span m="3422530">choose</span> <span m="3423230">two</span> <span m="3423490">elements</span> <span m="3424180">out</span> <span m="3424500">of</span> <span m="3424900">a</span> <span m="3425010">set</span> <span m="3425380">of</span> <span m="3425540">four</span> <span m="3427600">which</span> <span m="3427750">represent</span> <span m="3428270">all</span> <span m="3428380">the</span> <span m="3428480">suits.</span> <span m="3429260">So</span> <span m="3430520">that's</span> <span m="3430810">4</span> <span m="3431550">choose</span> <span m="3431920">2.</span> </p>
<p><span m="3439140">Then,</span> <span m="3439370">three,</span> <span m="3440540">we're</span> <span m="3440660">going</span> <span m="3440910">to</span> <span m="3441050">choose</span> <span m="3441350">the</span> <span m="3441460">value</span> <span m="3442600">of</span> <span m="3442990">the</span> <span m="3443370">second</span> <span m="3444130">pair.</span> <span m="3448340">Well,</span> <span m="3448490">that's</span> <span m="3448710">easy</span> <span m="3449040">because</span> <span m="3449910">the</span> <span m="3449990">second</span> <span m="3450380">pair,</span> <span m="3450580">by</span> <span m="3450680">definition,</span> <span m="3451600">must</span> <span m="3451870">have</span> <span m="3452020">a</span> <span m="3452150">different</span> <span m="3452690">value</span> <span m="3453720">than</span> <span m="3453890">the</span> <span m="3453990">one</span> <span m="3454190">in</span> <span m="3454250">the</span> <span m="3454330">first</span> <span m="3454720">pair.</span> <span m="3456050">Well,</span> <span m="3456570">I've</span> <span m="3456880">already</span> <span m="3457330">chosen</span> <span m="3457740">one value</span> <span m="3458330">so</span> <span m="3458590">I have</span> <span m="3458860">12</span> <span m="3459190">choices</span> <span m="3459660">left.</span> <span m="3461780">So</span> <span m="3462730">that's</span> <span m="3462960">no</span> <span m="3463030">problem.</span> </p>
<p><span m="3469870">And</span> <span m="3470770">now</span> <span m="3470920">we</span> <span m="3471020">can</span> <span m="3471190">continue</span> <span m="3471900">and</span> <span m="3472520">do</span> <span m="3472620">the</span> <span m="3472730">same</span> <span m="3473050">thing.</span> <span m="3473430">So</span> <span m="3473520">we're going</span> <span m="3473870">to</span> <span m="3473950">count</span> <span m="3474310">the</span> <span m="3474870">number</span> <span m="3475120">of</span> <span m="3475220">suites</span> <span m="3475470">that</span> <span m="3475740">are</span> <span m="3475850">possible</span> <span m="3476300">for the</span> <span m="3476620">second</span> <span m="3478080">pair.</span> <span m="3481310">And</span> <span m="3481630">it's</span> <span m="3481810">the</span> <span m="3481890">same</span> <span m="3483860">number</span> <span m="3484240">as</span> <span m="3484510">we</span> <span m="3484640">have</span> <span m="3484860">for</span> <span m="3484990">the</span> <span m="3485070">first</span> <span m="3485460">pair.</span> <span m="3486870">So</span> <span m="3487190">we</span> <span m="3487300">have</span> <span m="3487740">a</span> <span m="3487800">number</span> <span m="3488060">of</span> <span m="3488250">suits</span> <span m="3488730">of</span> <span m="3489010">the</span> <span m="3489290">second</span> <span m="3489560">pair.</span> <span m="3490820">Again,</span> <span m="3492120">we</span> <span m="3492240">need</span> <span m="3492450">to</span> <span m="3492540">choose</span> <span m="3492920">two</span> <span m="3493380">suits</span> <span m="3494240">out</span> <span m="3494490">of</span> <span m="3494630">the</span> <span m="3494800">complete</span> <span m="3495240">set</span> <span m="3495470">of</span> <span m="3495550">suits,</span> <span m="3496520">which</span> <span m="3496750">has</span> <span m="3497150">four</span> <span m="3497620">possibilities.</span> <span m="3498470">So it's</span> <span m="3498670">4</span> <span m="3498990">choose</span> <span m="3499480">2.</span> </p>
<p><span m="3500790">And</span> <span m="3501030">now</span> <span m="3501220">we</span> <span m="3501350">can</span> <span m="3501600">have</span> <span m="3503790">still</span> <span m="3504050">a</span> <span m="3504120">choice</span> <span m="3504500">for</span> <span m="3504630">the</span> <span m="3504730">last</span> <span m="3505090">cards.</span> <span m="3505870">We</span> <span m="3506010">have</span> <span m="3506330">now</span> <span m="3506880">two</span> <span m="3507070">pairs.</span> <span m="3507510">We</span> <span m="3507630">still</span> <span m="3507850">have</span> <span m="3508010">a</span> <span m="3508080">fifth</span> <span m="3508450">card.</span> <span m="3509220">The</span> <span m="3509430">fifth</span> <span m="3509760">card</span> <span m="3510260">also</span> <span m="3510580">has</span> <span m="3510790">a</span> <span m="3510840">value.</span> <span m="3513800">Actually,</span> <span m="3514170">I</span> <span m="3514210">did</span> <span m="3514420">not</span> <span m="3514840">write</span> <span m="3515130">it</span> <span m="3515280">down,</span> <span m="3515560">but</span> <span m="3516050">if we</span> <span m="3516500">talk</span> <span m="3516760">about</span> <span m="3517000">a</span> <span m="3517060">hand</span> <span m="3517470">with</span> <span m="3517730">two</span> <span m="3517950">pairs,</span> <span m="3518660">we</span> <span m="3518790">mean</span> <span m="3519060">two</span> <span m="3519260">cards</span> <span m="3519480">of</span> <span m="3519640">one</span> <span m="3519910">value,</span> <span m="3520750">two</span> <span m="3520970">cards of</span> <span m="3521440">another</span> <span m="3521740">value,</span> <span m="3522640">and</span> <span m="3522820">the</span> <span m="3522890">fifth card--</span> <span m="3523220">the</span> <span m="3523550">extra</span> <span m="3523920">card--</span> <span m="3524980">has</span> <span m="3525180">yet</span> <span m="3525500">another</span> <span m="3525830">value.</span> <span m="3527000">Because</span> <span m="3527230">otherwise</span> <span m="3527570">you</span> <span m="3527680">would</span> <span m="3527820">have</span> <span m="3528000">a</span> <span m="3528050">full</span> <span m="3528300">house</span> <span m="3529370">and we</span> <span m="3529550">have already</span> <span m="3529900">counted</span> <span m="3530210">those.</span> </p>
<p><span m="3532760">OK.</span> <span m="3533110">So</span> <span m="3533550">value</span> <span m="3533850">of</span> <span m="3533980">the</span> <span m="3534140">extra</span> <span m="3535190">card.</span> <span m="3538140">Well,</span> <span m="3538400">I've</span> <span m="3538630">chosen</span> <span m="3539050">already</span> <span m="3539340">two</span> <span m="3539610">values.</span> <span m="3540230">The</span> <span m="3540340">one</span> <span m="3541130">for</span> <span m="3541440">the</span> <span m="3541520">first</span> <span m="3541850">pair, the one</span> <span m="3542250">of</span> <span m="3542360">the</span> <span m="3542440">second</span> <span m="3542790">pair,</span> <span m="3543110">so</span> <span m="3543280">there</span> <span m="3543530">are</span> <span m="3543750">11</span> <span m="3544120">choices</span> <span m="3544610">left.</span> <span m="3545830">And</span> <span m="3546010">finally,</span> <span m="3547290">I</span> <span m="3547420">can</span> <span m="3547610">choose</span> <span m="3547870">a</span> <span m="3547980">suit</span> <span m="3548500">of</span> <span m="3548740">the</span> <span m="3548880">extra</span> <span m="3549270">card.</span> <span m="3550390">Well,</span> <span m="3550650">I</span> <span m="3550850">have</span> <span m="3552570">one</span> <span m="3553150">out of four</span> <span m="3553570">choices</span> <span m="3554350">for</span> <span m="3554490">my</span> <span m="3554630">suit.</span> </p>
<p><span m="3556320">So</span> <span m="3557770">again</span> <span m="3558070">we</span> <span m="3558160">can</span> <span m="3558280">use</span> <span m="3558460">the</span> <span m="3558530">generalized</span> <span m="3559010">product</span> <span m="3559420">rule</span> <span m="3559900">and</span> <span m="3560120">we</span> <span m="3560260">can</span> <span m="3560440">say</span> <span m="3560820">that</span> <span m="3561460">we</span> <span m="3561650">have</span> <span m="3562900">30</span> <span m="3563360">choices</span> <span m="3563900">for</span> <span m="3564080">my</span> <span m="3564240">first</span> <span m="3566410">entry</span> <span m="3566690">in my</span> <span m="3567160">presentation.</span> <span m="3568650">Right?</span> <span m="3569000">For</span> <span m="3571490">the</span> <span m="3571570">first</span> <span m="3572010">choice.</span> <span m="3572980">Then</span> <span m="3573190">given</span> <span m="3573430">the</span> <span m="3573510">first</span> <span m="3573830">choice,</span> <span m="3574060">I</span> <span m="3574190">have</span> <span m="3574400">4</span> <span m="3574680">choose</span> <span m="3574970">2</span> <span m="3575510">choices</span> <span m="3575910">for</span> <span m="3576010">the</span> <span m="3576100">second.</span> <span m="3577190">And</span> <span m="3577370">then</span> <span m="3578240">12</span> <span m="3578600">choices</span> <span m="3578990">for</span> <span m="3579150">the</span> <span m="3579250">third</span> <span m="3579936">if</span> <span m="3580280">I've already</span> <span m="3580410">chosen</span> <span m="3580800">the</span> <span m="3580870">first</span> <span m="3581250">two.</span> <span m="3581580">And</span> <span m="3581790">so</span> <span m="3582040">on.</span> </p>
<p><span m="3582750">So</span> <span m="3582960">I</span> <span m="3583020">can</span> <span m="3583150">use</span> <span m="3583350">the</span> <span m="3583400">generalized</span> <span m="3583950">product</span> <span m="3584270">rule</span> <span m="3587090">and</span> <span m="3587590">count</span> <span m="3587970">these</span> <span m="3589430">representations.</span> <span m="3590520">So</span> <span m="3590630">the</span> <span m="3590750">number</span> <span m="3592106">of</span> <span m="3592510">representations</span> <span m="3594320">is</span> <span m="3594530">actually</span> <span m="3594980">equal</span> <span m="3595330">to</span> <span m="3595480">13</span> <span m="3596540">times</span> <span m="3597460">4</span> <span m="3598060">choose</span> <span m="3598410">2</span> <span m="3599200">times</span> <span m="3600130">12</span> <span m="3600800">times</span> <span m="3601720">4</span> <span m="3602050">choose</span> <span m="3603020">2</span> <span m="3603240">again</span> <span m="3604370">times</span> <span m="3604630">11</span> <span m="3605170">times</span> <span m="3606210">4</span> <span m="3606570">choose</span> <span m="3606760">1.</span> </p>
<p><span m="3610130">So</span> <span m="3610790">is</span> <span m="3611010">this</span> <span m="3611260">a</span> <span m="3611330">number</span> <span m="3612480">of</span> <span m="3612820">the</span> <span m="3612900">hands</span> <span m="3613280">with</span> <span m="3613440">two</span> <span m="3613710">pairs?</span> <span m="3615230">So</span> <span m="3615400">this</span> <span m="3615570">seems</span> <span m="3615790">to</span> <span m="3615870">be</span> <span m="3616030">pretty</span> <span m="3616310">reasonable.</span> <span m="3616930">But</span> <span m="3617580">can</span> <span m="3617740">you</span> <span m="3617840">see</span> <span m="3620140">something</span> <span m="3620470">that</span> <span m="3620600">has</span> <span m="3620760">happened</span> <span m="3621180">here</span> <span m="3621710">that</span> <span m="3621870">we</span> <span m="3622420">also</span> <span m="3623780">saw</span> <span m="3624300">actually</span> <span m="3624580">last</span> <span m="3624880">lecture?</span> <span m="3627810">Like,</span> <span m="3627970">do</span> <span m="3628100">I</span> <span m="3628200">know</span> <span m="3628670">for</span> <span m="3628860">sure</span> <span m="3629260">that</span> <span m="3630330">the</span> <span m="3630410">hands</span> <span m="3630700">with</span> <span m="3630830">two</span> <span m="3631230">pairs</span> <span m="3632520">is--</span> <span m="3633570">well,</span> <span m="3634240">that</span> <span m="3634640">are</span> <span m="3634720">this</span> <span m="3634910">number</span> <span m="3635940">of</span> <span m="3636280">hands</span> <span m="3636610">with</span> <span m="3636750">two</span> <span m="3636960">pairs.</span> <span m="3638820">What</span> <span m="3638950">do</span> <span m="3639050">I</span> <span m="3639130">need</span> <span m="3639340">to</span> <span m="3639480">check</span> <span m="3640020">in</span> <span m="3640180">order</span> <span m="3640320">to</span> <span m="3641280">make</span> <span m="3641470">sure</span> <span m="3641760">that that</span> <span m="3641950">is</span> <span m="3642110">true?</span> </p>
<p><span m="3643100">Well,</span> <span m="3644000">I</span> <span m="3644090">need</span> <span m="3644250">to</span> <span m="3644350">make--</span> <span m="3646480">I</span> <span m="3646560">need</span> <span m="3646700">to</span> <span m="3646790">prove,</span> <span m="3647150">essentially,</span> <span m="3647800">that</span> <span m="3648340">this</span> <span m="3648540">representation</span> <span m="3648920">is</span> <span m="3649300">actually</span> <span m="3649690">bijection</span> <span m="3650550">from</span> <span m="3650810">the</span> <span m="3650890">hands with</span> <span m="3651240">two</span> <span m="3651530">pairs</span> <span m="3652340">to</span> <span m="3653250">this</span> <span m="3653490">types</span> <span m="3654200">of</span> <span m="3654560">sequences.</span> <span m="3655470">So</span> <span m="3655520">is</span> <span m="3655650">that</span> <span m="3655820">true?</span> <span m="3657740">Or</span> <span m="3657960">does</span> <span m="3658350">this</span> <span m="3658880">mapping</span> <span m="3659200">have</span> <span m="3659770">a</span> <span m="3659880">different</span> <span m="3660230">property?</span> <span m="3663100">Any</span> <span m="3664972">ideas?</span> <span m="3667011">Yeah?</span> <span m="3667488">Over</span> <span m="3667965">there.</span> </p>
<p><span m="3669396">AUDIENCE: No.</span> </p>
<p><span m="3670350">PROFESSOR: No?</span> <span m="3670830">You</span> <span m="3671225">don't want</span> <span m="3671620">to answer?</span> <span m="3672505">OK.</span> <span m="3673990">Well,</span> <span m="3676260">let</span> <span m="3676460">me--</span> <span m="3678420">when</span> <span m="3678920">we</span> <span m="3679050">talked</span> <span m="3679360">about</span> <span m="3680270">the</span> <span m="3680570">chess</span> <span m="3680890">game</span> <span m="3681190">last</span> <span m="3681520">time,</span> <span m="3682270">or</span> <span m="3682600">with</span> <span m="3683160">the</span> <span m="3683300">rooks,</span> <span m="3684360">we</span> <span m="3684490">could</span> <span m="3685150">essentially</span> <span m="3685620">had</span> <span m="3685810">two</span> <span m="3686080">rooks</span> <span m="3686360">that</span> <span m="3686460">we</span> <span m="3686560">could</span> <span m="3686720">choose</span> <span m="3687820">and</span> <span m="3688460">put</span> <span m="3688600">them</span> <span m="3688800">on</span> <span m="3688980">different</span> <span m="3689440">positions</span> <span m="3690330">with</span> <span m="3690560">no</span> <span m="3691670">shared</span> <span m="3692010">rows</span> <span m="3692240">or</span> <span m="3692390">shared</span> <span m="3692730">columns.</span> <span m="3693950">And</span> <span m="3694120">here</span> <span m="3694290">we</span> <span m="3694410">have</span> <span m="3695500">two</span> <span m="3695960">pairs</span> <span m="3696700">with</span> <span m="3697150">no</span> <span m="3697450">shared</span> <span m="3698050">suit--</span> <span m="3698900">with</span> <span m="3699170">no</span> <span m="3699370">shared</span> <span m="3699780">values.</span> </p>
<p><span m="3701080">But</span> <span m="3701180">what</span> <span m="3701210">can</span> <span m="3701380">we</span> <span m="3701500">do?</span> <span m="3702640">We</span> <span m="3703250">can</span> <span m="3703470">actually</span> <span m="3704090">interchange</span> <span m="3704755">the</span> <span m="3705100">first</span> <span m="3705450">and</span> <span m="3705530">the</span> <span m="3705590">second</span> <span m="3706030">pair.</span> <span m="3706850">So</span> <span m="3707780">as</span> <span m="3707990">an</span> <span m="3708070">example,</span> <span m="3711780">we</span> <span m="3711990">can</span> <span m="3712200">have</span> <span m="3712650">two</span> <span m="3713280">representations</span> <span m="3714340">that</span> <span m="3714520">map</span> <span m="3714770">to</span> <span m="3714900">the</span> <span m="3715000">same</span> <span m="3715330">hand.</span> <span m="3715730">So let</span> <span m="3715830">me</span> <span m="3715910">give</span> <span m="3716060">an</span> <span m="3716150">example.</span> </p>
<p><span m="3716880">So</span> <span m="3717000">for</span> <span m="3717140">example,</span> <span m="3717600">suppose</span> <span m="3718060">you</span> <span m="3718190">choose</span> <span m="3718820">the</span> <span m="3718980">value</span> <span m="3719300">3</span> <span m="3719670">for</span> <span m="3719870">the</span> <span m="3719960">first</span> <span m="3720410">pair</span> <span m="3721480">and</span> <span m="3721800">then</span> <span m="3722510">I'm</span> <span m="3722810">choosing</span> <span m="3723720">the</span> <span m="3723940">set</span> <span m="3725110">diamonds</span> <span m="3725710">and</span> <span m="3725830">clubs</span> <span m="3726390">for</span> <span m="3726970">the</span> <span m="3727090">suits</span> <span m="3727550">of</span> <span m="3727680">those.</span> <span m="3728730">And</span> <span m="3728840">say</span> <span m="3729320">the</span> <span m="3729410">second</span> <span m="3729760">pair</span> <span m="3730410">is</span> <span m="3730900">a</span> <span m="3731020">queen</span> <span m="3732190">of</span> <span m="3732350">diamonds</span> <span m="3733630">and</span> <span m="3733840">hearts.</span> <span m="3735020">And</span> <span m="3735330">finally,</span> <span m="3735820">I</span> <span m="3735940">have</span> <span m="3736410">an</span> <span m="3736610">ace</span> <span m="3737250">of</span> <span m="3737410">clubs.</span> <span m="3738370">Well,</span> <span m="3738490">this</span> <span m="3738670">would</span> <span m="3738780">be</span> <span m="3738940">a</span> <span m="3738970">proper</span> <span m="3741060">sequence</span> <span m="3741740">according</span> <span m="3742090">to</span> <span m="3742210">this</span> <span m="3742390">representation.</span> </p>
<p><span m="3744070">But</span> <span m="3744220">this</span> <span m="3744430">maps</span> <span m="3744800">to</span> <span m="3744920">a</span> <span m="3745050">hand</span> <span m="3745580">that</span> <span m="3745720">does</span> <span m="3745890">also</span> <span m="3746210">map</span> <span m="3746540">to</span> <span m="3746830">if</span> <span m="3746930">we</span> <span m="3747710">interchange</span> <span m="3748490">the</span> <span m="3748590">first</span> <span m="3748930">and the</span> <span m="3749060">second</span> <span m="3749430">pair.</span> <span m="3750540">We</span> <span m="3750640">can</span> <span m="3750820">also</span> <span m="3751070">start</span> <span m="3751430">off</span> <span m="3751720">with</span> <span m="3753130">the</span> <span m="3753270">queen</span> <span m="3753610">for</span> <span m="3753800">the</span> <span m="3753880">first</span> <span m="3754290">pair</span> <span m="3756020">and</span> <span m="3756170">we</span> <span m="3756290">have</span> <span m="3756530">diamonds</span> <span m="3757050">and</span> <span m="3757210">hearts.</span> <span m="3758500">And</span> <span m="3758690">we</span> <span m="3758780">have</span> <span m="3759210">three</span> <span m="3760460">for</span> <span m="3760570">the</span> <span m="3760650">second</span> <span m="3761010">pair</span> <span m="3762080">with</span> <span m="3763450">diamonds</span> <span m="3763880">and</span> <span m="3764110">clubs.</span> <span m="3767440">And</span> <span m="3767620">then</span> <span m="3767740">we</span> <span m="3767870">finally</span> <span m="3768360">have</span> <span m="3769090">the</span> <span m="3769200">fifth</span> <span m="3769550">card</span> <span m="3769830">which</span> <span m="3770110">is the</span> <span m="3770220">same,</span> <span m="3770700">the</span> <span m="3770820">ace</span> <span m="3771080">of</span> <span m="3771210">clubs.</span> </p>
<p><span m="3772820">Now</span> <span m="3772970">these</span> <span m="3773210">map</span> <span m="3774180">to</span> <span m="3774690">exactly</span> <span m="3775310">the</span> <span m="3775400">same</span> <span m="3776190">hand.</span> <span m="3777730">We</span> <span m="3778150">just</span> <span m="3778380">need</span> <span m="3778500">to</span> <span m="3778570">change</span> <span m="3778900">the</span> <span m="3779000">first</span> <span m="3779290">and the</span> <span m="3779400">second</span> <span m="3779740">pair.</span> <span m="3780495">Now</span> <span m="3780890">this</span> <span m="3781090">is a</span> <span m="3781320">kind</span> <span m="3781650">of</span> <span m="3782910">problem</span> <span m="3783710">or</span> <span m="3785510">a</span> <span m="3785570">mental</span> <span m="3786630">sort</span> <span m="3786960">of</span> <span m="3789350">confusion</span> <span m="3790020">that</span> <span m="3790620">very</span> <span m="3790970">easily</span> <span m="3791290">happened.</span> <span m="3791760">This</span> <span m="3791940">is</span> <span m="3792210">a</span> <span m="3792280">rather</span> <span m="3792630">easy</span> <span m="3793090">example</span> <span m="3793570">where</span> <span m="3793730">you</span> <span m="3793810">can</span> <span m="3793970">see</span> <span m="3794310">it.</span> <span m="3794920">But</span> <span m="3795100">if</span> <span m="3795230">you do</span> <span m="3795500">counting,</span> <span m="3797510">we</span> <span m="3797620">really</span> <span m="3797860">have</span> <span m="3798020">to</span> <span m="3798160">take</span> <span m="3798460">care</span> <span m="3799470">that</span> <span m="3800340">we</span> <span m="3800560">make</span> <span m="3800800">sure</span> <span m="3801650">that</span> <span m="3802180">each</span> <span m="3802660">of</span> <span m="3803030">the</span> <span m="3804950">sequences</span> <span m="3805680">in</span> <span m="3805810">this</span> <span m="3805980">representation</span> <span m="3806920">are</span> <span m="3807250">really</span> <span m="3807500">represented</span> <span m="3809120">by,</span> <span m="3809280">say,</span> <span m="3809560">one</span> <span m="3811410">hand,</span> <span m="3811980">in</span> <span m="3812040">this</span> <span m="3812220">case,</span> <span m="3812450">with</span> <span m="3812650">two</span> <span m="3812940">pairs.</span> </p>
<p><span m="3814000">But</span> <span m="3814140">how</span> <span m="3814330">can</span> <span m="3814470">we</span> <span m="3814570">remedy</span> <span m="3814990">this?</span> <span m="3816430">It</span> <span m="3816600">turns</span> <span m="3816950">that</span> <span m="3817230">we</span> <span m="3817340">can--</span> <span m="3818630">by</span> <span m="3818890">interchanging</span> <span m="3819800">these</span> <span m="3820010">two</span> <span m="3820250">pairs,</span> <span m="3822570">these</span> <span m="3822880">two</span> <span m="3823400">sequences</span> <span m="3824190">are</span> <span m="3824430">the</span> <span m="3824750">only</span> <span m="3825150">two</span> <span m="3825380">sequences</span> <span m="3826000">that</span> <span m="3826160">map</span> <span m="3826360">to</span> <span m="3826470">the</span> <span m="3826570">same</span> <span m="3826860">hand.</span> <span m="3827990">So</span> <span m="3828200">we</span> <span m="3828320">know</span> <span m="3828630">that</span> <span m="3828760">this</span> <span m="3828940">is</span> <span m="3829070">not</span> <span m="3829220">a</span> <span m="3829410">bijection</span> <span m="3830200">but</span> <span m="3830400">it is</span> <span m="3830640">a</span> <span m="3830770">2-to-1</span> <span m="3832130">mapping.</span> <span m="3834090">So</span> <span m="3834300">now</span> <span m="3834600">we</span> <span m="3834740">can</span> <span m="3835480">choose--</span> <span m="3836750">we</span> <span m="3836790">can</span> <span m="3836930">use</span> <span m="3837230">the</span> <span m="3837730">division</span> <span m="3838370">rule.</span> </p>
<p><span m="3839590">And</span> <span m="3840960">by</span> <span m="3841120">the</span> <span m="3841230">division</span> <span m="3841790">rule</span> <span m="3844770">we</span> <span m="3844890">now</span> <span m="3845120">know</span> <span m="3846220">that</span> <span m="3846430">the</span> <span m="3846530">number</span> <span m="3847000">of</span> <span m="3848930">hands with</span> <span m="3849210">two</span> <span m="3849470">pairs</span> <span m="3854530">is</span> <span m="3854820">actually</span> <span m="3855340">equal</span> <span m="3855860">to</span> <span m="3856000">this</span> <span m="3856210">whole</span> <span m="3856450">thing</span> <span m="3856690">over</span> <span m="3856980">here</span> <span m="3858620">divided</span> <span m="3859210">by</span> <span m="3859370">2</span> <span m="3860070">because</span> <span m="3860370">we</span> <span m="3860490">have a</span> <span m="3860600">2-to-1</span> <span m="3861170">function.</span> <span m="3863448">Now,</span> <span m="3863900">this</span> <span m="3864110">is</span> <span m="3864550">really</span> <span m="3865290">something</span> <span m="3865690">that</span> <span m="3865850">is</span> <span m="3865890">pretty</span> <span m="3866200">disturbing</span> <span m="3866800">because</span> <span m="3867040">we</span> <span m="3867140">have</span> <span m="3867310">a</span> <span m="3867350">near</span> <span m="3867780">miss</span> <span m="3868220">in</span> <span m="3868450">our</span> <span m="3869060">reasoning.</span> <span m="3872770">And</span> <span m="3872960">therefore,</span> <span m="3873550">it is</span> <span m="3873780">important</span> <span m="3874200">to</span> <span m="3874300">keep</span> <span m="3874510">in</span> <span m="3874610">mind</span> <span m="3875530">the</span> <span m="3875610">following</span> <span m="3876030">guidelines</span> <span m="3877170">when</span> <span m="3877560">we</span> <span m="3877750">do</span> <span m="3877910">counting.</span> </p>
<p><span m="3884170">It's</span> <span m="3884720">very</span> <span m="3884980">important</span> <span m="3885285">to,</span> <span m="3885590">first</span> <span m="3885730">of</span> <span m="3885960">all,</span> <span m="3886330">check</span> <span m="3887230">whether</span> <span m="3887520">it's</span> <span m="3887710">truly</span> <span m="3888060">a</span> <span m="3888270">bijection.</span> <span m="3889060">And</span> <span m="3889990">you</span> <span m="3890090">need</span> <span m="3890270">to</span> <span m="3890380">know</span> <span m="3890640">how</span> <span m="3890840">many</span> <span m="3892430">sequences</span> <span m="3893080">or</span> <span m="3893190">maps</span> <span m="3893550">to</span> <span m="3893640">the</span> <span m="3893760">same--</span> <span m="3895970">to</span> <span m="3896070">the</span> <span m="3896080">same</span> <span m="3896350">hand.</span> <span m="3897350">So</span> <span m="3898110">the</span> <span m="3898200">guidelines</span> <span m="3898850">are</span> <span m="3900000">as</span> <span m="3900220">follows.</span> <span m="3901612">They're</span> <span m="3901970">pretty</span> <span m="3902230">straightforward</span> <span m="3903100">actually.</span> </p>
<p><span m="3905330">First</span> <span m="3905790">of</span> <span m="3905910">all,</span> <span m="3907350">if</span> <span m="3907590">we</span> <span m="3907730">have</span> <span m="3908590">a</span> <span m="3908890">function</span> <span m="3909420">f</span> <span m="3909840">that</span> <span m="3910080">maps</span> <span m="3910630">from</span> <span m="3910950">A</span> <span m="3911200">to</span> <span m="3911470">B,</span> <span m="3912760">then</span> <span m="3913820">we</span> <span m="3913970">would</span> <span m="3914170">really</span> <span m="3914470">like</span> <span m="3914700">to</span> <span m="3914910">check</span> <span m="3915410">very</span> <span m="3915680">carefully</span> <span m="3917310">whether</span> <span m="3917720">the</span> <span m="3918010">number--</span> <span m="3919100">what</span> <span m="3919250">is</span> <span m="3919410">the</span> <span m="3919560">number</span> <span m="3919860">of</span> <span m="3920090">elements</span> <span m="3922350">of</span> <span m="3922670">A</span> <span m="3924320">that</span> <span m="3924490">are</span> <span m="3924580">mapped</span> <span m="3928040">to</span> <span m="3929880">each</span> <span m="3934750">element</span> <span m="3936750">of</span> <span m="3937030">B?</span> <span m="3938380">So</span> <span m="3939110">we</span> <span m="3939260">check</span> <span m="3939800">how</span> <span m="3939990">many</span> <span m="3940320">to</span> <span m="3940490">one</span> <span m="3940980">of a</span> <span m="3941220">mapping</span> <span m="3941570">this</span> <span m="3941780">really</span> <span m="3942050">is.</span> <span m="3943080">And</span> <span m="3943530">after</span> <span m="3943790">this</span> <span m="3944570">we</span> <span m="3944720">will</span> <span m="3944860">then</span> <span m="3945220">apply</span> <span m="3946960">the</span> <span m="3947070">division</span> <span m="3947490">rule.</span> </p>
<p><span m="3951710">It's</span> <span m="3951830">very</span> <span m="3952020">important</span> <span m="3952350">to</span> <span m="3952460">check</span> <span m="3952770">this.</span> <span m="3953870">But</span> <span m="3954460">very</span> <span m="3954650">often,</span> <span m="3956050">we're</span> <span m="3956200">making</span> <span m="3956490">mistakes.</span> <span m="3956980">For</span> <span m="3957160">example,</span> <span m="3957310">right</span> <span m="3957530">now</span> <span m="3957790">I'm</span> <span m="3957910">doing</span> <span m="3958130">some</span> <span m="3958310">research</span> <span m="3958920">and</span> <span m="3959380">I've</span> <span m="3959510">been</span> <span m="3959740">counting</span> <span m="3960110">something.</span> <span m="3961030">Turns</span> <span m="3961380">out</span> <span m="3961570">it--</span> <span m="3965625">if</span> <span m="3965910">I</span> <span m="3965950">use</span> <span m="3966170">a</span> <span m="3966230">different</span> <span m="3966570">method</span> <span m="3966990">to</span> <span m="3967110">count</span> <span m="3967390">the same</span> <span m="3967680">thing</span> <span m="3967880">I</span> <span m="3967920">get</span> <span m="3968100">a</span> <span m="3968140">different</span> <span m="3968430">answer.</span> <span m="3968730">So</span> <span m="3968920">I made</span> <span m="3969120">a</span> <span m="3969160">mistake</span> <span m="3969570">somewhere.</span> </p>
<p><span m="3970820">So</span> <span m="3972190">that's</span> <span m="3972460">the</span> <span m="3972550">second</span> <span m="3972920">guideline.</span> <span m="3973590">So</span> <span m="3973870">what</span> <span m="3974040">you</span> <span m="3974140">want</span> <span m="3974330">to</span> <span m="3974450">do</span> <span m="3974700">is</span> <span m="3974900">you</span> <span m="3975130">want</span> <span m="3975340">to</span> <span m="3976320">try</span> <span m="3977360">and</span> <span m="3977580">solving</span> <span m="3977980">a</span> <span m="3978060">problem</span> <span m="3978660">in</span> <span m="3978790">multiple</span> <span m="3979240">ways.</span> <span m="3983420">So</span> <span m="3983590">try</span> <span m="3983860">solving</span> <span m="3984250">a</span> <span m="3984330">problem</span> <span m="3986130">in</span> <span m="3986350">a</span> <span m="3986400">different</span> <span m="3986850">way.</span> <span m="3987680">And</span> <span m="3987840">especially</span> <span m="3988360">with</span> <span m="3988530">counting</span> <span m="3988980">this</span> <span m="3989150">is</span> <span m="3989270">very</span> <span m="3989520">helpful.</span> <span m="3993290">So</span> <span m="3994680">this</span> <span m="3994950">will</span> <span m="3995070">lead</span> <span m="3995260">to</span> <span m="3995370">an</span> <span m="3995460">extra</span> <span m="3995780">check.</span> <span m="3996420">So</span> <span m="3996690">we</span> <span m="3996830">will</span> <span m="3996950">do</span> <span m="3997140">that</span> <span m="3997440">for</span> <span m="3997560">this</span> <span m="3997730">particular</span> <span m="3998210">example.</span> <span m="4002340">And</span> <span m="4002480">it</span> <span m="4002560">generally</span> <span m="4003100">would</span> <span m="4003490">always</span> <span m="4003950">like</span> <span m="4004270">to</span> <span m="4005030">find</span> <span m="4006240">multiple</span> <span m="4006770">ways</span> <span m="4007750">to</span> <span m="4007960">prove</span> <span m="4008380">things,</span> <span m="4009040">I</span> <span m="4009310">would</span> <span m="4009500">say,</span> <span m="4009780">because</span> <span m="4010790">at</span> <span m="4010920">least</span> <span m="4011150">you</span> <span m="4011240">want</span> <span m="4011400">to</span> <span m="4011540">have</span> <span m="4012020">multiple--</span> <span m="4014680">maybe</span> <span m="4015090">not</span> <span m="4015160">a complete</span> <span m="4015760">proof</span> <span m="4016110">but</span> <span m="4016280">subparts</span> <span m="4017130">of</span> <span m="4017270">proofs--</span> <span m="4017630">you</span> <span m="4017730">may</span> <span m="4017900">want</span> <span m="4018050">to</span> <span m="4018130">find</span> <span m="4018370">different</span> <span m="4018690">ways</span> <span m="4018970">why</span> <span m="4019120">that</span> <span m="4019290">is</span> <span m="4019450">true.</span> </p>
<p><span m="4020480">So</span> <span m="4020520">that's</span> <span m="4020730">how</span> <span m="4020890">I</span> <span m="4021040">usually</span> <span m="4021420">do</span> <span m="4023230">my</span> <span m="4023380">proofing</span> <span m="4023880">of</span> <span m="4024040">things.</span> <span m="4025380">So</span> <span m="4026510">let's</span> <span m="4027630">find</span> <span m="4028000">a</span> <span m="4028070">second</span> <span m="4028520">way</span> <span m="4028800">to</span> <span m="4028950">count</span> <span m="4029240">this</span> <span m="4030010">over</span> <span m="4030290">here,</span> <span m="4031370">to</span> <span m="4031460">do</span> <span m="4031600">the</span> <span m="4031700">double</span> <span m="4031990">check.</span> <span m="4033720">So</span> <span m="4033920">we're</span> <span m="4034040">going</span> <span m="4034310">to</span> <span m="4035230">create</span> <span m="4035740">a</span> <span m="4035940">different</span> <span m="4036360">representation</span> <span m="4038050">that</span> <span m="4038220">should</span> <span m="4038720">actually</span> <span m="4040450">lead</span> <span m="4040980">to</span> <span m="4041610">a</span> <span m="4042060">bijection.</span> <span m="4042750">So</span> <span m="4043010">I</span> <span m="4043090">want</span> <span m="4043280">to</span> <span m="4043410">find</span> <span m="4043700">a</span> <span m="4043770">representation</span> <span m="4044560">that's a</span> <span m="4044790">bijection,</span> <span m="4045220">a</span> <span m="4045310">1-to-1</span> <span m="4046540">mapping.</span> </p>
<p><span m="4049020">So</span> <span m="4049230">how</span> <span m="4049420">do</span> <span m="4049530">we</span> <span m="4049650">do</span> <span m="4049840">this?</span> <span m="4051730">I</span> <span m="4051890">can--</span> <span m="4059240">let me</span> <span m="4059540">see where</span> <span m="4059800">I am.</span> <span m="4061290">OK.</span> <span m="4061590">So</span> <span m="4062860">what</span> <span m="4063020">I</span> <span m="4063120">could</span> <span m="4063440">do</span> <span m="4063710">is,</span> <span m="4067010">first</span> <span m="4067330">of</span> <span m="4067510">all,</span> <span m="4068470">I</span> <span m="4068610">can</span> <span m="4068840">choose</span> <span m="4069230">the</span> <span m="4069460">values</span> <span m="4071940">of</span> <span m="4072340">the</span> <span m="4072490">two</span> <span m="4072780">pairs.</span> <span m="4074260">So</span> <span m="4074750">in</span> <span m="4074900">this</span> <span m="4075140">case,</span> <span m="4076220">I</span> <span m="4076420">have</span> <span m="4076650">the</span> <span m="4076810">values</span> <span m="4077550">3</span> <span m="4077920">and</span> <span m="4078060">the</span> <span m="4078150">queen.</span> <span m="4079220">So</span> <span m="4079430">how</span> <span m="4079580">many</span> <span m="4079860">choices</span> <span m="4080270">do</span> <span m="4080430">I</span> <span m="4080550">have?</span> <span m="4081700">Well,</span> <span m="4081810">I</span> <span m="4081870">can</span> <span m="4082080">choose</span> <span m="4082460">two</span> <span m="4082725">out</span> <span m="4082990">of</span> <span m="4083190">13</span> <span m="4083650">choices.</span> </p>
<p><span m="4086790">Secondly,</span> <span m="4088330">I'm</span> <span m="4088550">going</span> <span m="4088730">to</span> <span m="4088840">describe</span> <span m="4089280">the</span> <span m="4089400">suit</span> <span m="4090040">of</span> <span m="4090590">the</span> <span m="4090740">smaller</span> <span m="4091340">pair</span> <span m="4092950">which</span> <span m="4093120">is</span> <span m="4093180">uniquely</span> <span m="4093700">defined</span> <span m="4094160">because</span> <span m="4094490">they</span> <span m="4094590">have</span> <span m="4094850">two</span> <span m="4095090">values.</span> <span m="4095920">One</span> <span m="4096210">is</span> <span m="4096300">smaller</span> <span m="4097270">and</span> <span m="4097420">the</span> <span m="4097460">suit</span> <span m="4097729">of the</span> <span m="4097810">smaller</span> <span m="4098430">pair</span> <span m="4099410">I</span> <span m="4099540">can</span> <span m="4099750">choose</span> <span m="4100460">by</span> <span m="4101560">taking</span> <span m="4102069">one</span> <span m="4102430">out</span> <span m="4102630">the</span> <span m="4102729">four--</span> <span m="4105470">two</span> <span m="4105779">out</span> <span m="4105930">of</span> <span m="4106020">four</span> <span m="4107500">choices</span> <span m="4107960">because</span> <span m="4108200">I</span> <span m="4108279">need</span> <span m="4108450">to</span> <span m="4109310">choose</span> <span m="4109660">two</span> <span m="4109990">suits</span> <span m="4110950">for</span> <span m="4111180">the</span> <span m="4111260">smaller</span> <span m="4111620">pair.</span> </p>
<p><span m="4112950">For</span> <span m="4113180">the</span> <span m="4113290">larger</span> <span m="4113500">pair,</span> <span m="4114050">I</span> <span m="4114160">do</span> <span m="4114300">the</span> <span m="4114410">same.</span> <span m="4118020">And</span> <span m="4118460">I</span> <span m="4118520">get</span> <span m="4119120">4</span> <span m="4119850">choose</span> <span m="4120200">2.</span> <span m="4121680">Now,</span> <span m="4122010">again,</span> <span m="4122319">I</span> <span m="4122430">have</span> <span m="4122700">the</span> <span m="4122800">value</span> <span m="4124540">of</span> <span m="4124800">the</span> <span m="4124960">extra</span> <span m="4125448">card.</span> <span m="4129359">And</span> <span m="4130160">I</span> <span m="4130290">have</span> <span m="4130500">11</span> <span m="4130979">choices.</span> <span m="4131399">I have</span> <span m="4131630">already</span> <span m="4132529">chosen</span> <span m="4132970">two</span> <span m="4133229">so</span> <span m="4133689">there are</span> <span m="4134120">11</span> <span m="4134450">choices</span> <span m="4134890">left.</span> <span m="4136090">And</span> <span m="4136460">finally,</span> <span m="4137560">the</span> <span m="4137700">suit</span> <span m="4138189">for</span> <span m="4139609">the</span> <span m="4139800">extra</span> <span m="4140140">card</span> <span m="4140600">can</span> <span m="4140790">be</span> <span m="4140899">done</span> <span m="4141300">in</span> <span m="4141670">4</span> <span m="4141970">choose</span> <span m="4142279">1</span> <span m="4142520">ways.</span> </p>
<p><span m="4143660">So</span> <span m="4144229">if</span> <span m="4144399">you</span> <span m="4144520">multiply</span> <span m="4145109">those</span> <span m="4145410">together,</span> <span m="4146290">we</span> <span m="4146380">get</span> <span m="4146710">13</span> <span m="4147810">times</span> <span m="4148609">12</span> <span m="4149160">divided</span> <span m="4149640">by</span> <span m="4149790">2.</span> <span m="4150990">And</span> <span m="4151200">then</span> <span m="4151220">we</span> <span m="4151479">get</span> <span m="4151680">times</span> <span m="4153319">4</span> <span m="4153620">choose</span> <span m="4153930">2</span> <span m="4154290">times</span> <span m="4154649">4</span> <span m="4155040">choose</span> <span m="4155450">2</span> <span m="4156240">times</span> <span m="4156670">11</span> <span m="4157930">times</span> <span m="4158310">4</span> <span m="4158550">choose</span> <span m="4158960">1.</span> <span m="4159779">And</span> <span m="4160300">we</span> <span m="4160439">can</span> <span m="4160630">see</span> <span m="4160950">that</span> <span m="4161130">that's</span> <span m="4161350">exactly</span> <span m="4161819">the</span> <span m="4161939">same</span> <span m="4162359">as</span> <span m="4162689">this</span> <span m="4162890">whole</span> <span m="4163210">product</span> <span m="4163710">divided</span> <span m="4164140">by</span> <span m="4164300">2.</span> <span m="4165390">We</span> <span m="4165540">actually</span> <span m="4165979">have</span> <span m="4166670">a</span> <span m="4166960">bijection.</span> <span m="4167490">This</span> <span m="4167689">particular</span> <span m="4168109">representation</span> <span m="4168484">is</span> <span m="4168859">a</span> <span m="4169029">1-to-1</span> <span m="4169590">mapping.</span> <span m="4170870">So</span> <span m="4170979">it's</span> <span m="4171120">always</span> <span m="4171430">good</span> <span m="4171600">to</span> <span m="4171830">find</span> <span m="4172609">a</span> <span m="4172670">second</span> <span m="4173120">way</span> <span m="4173290">to</span> <span m="4173420">prove</span> <span m="4173680">the</span> <span m="4173790">same</span> <span m="4177359">result.</span> </p>
<p><span m="4190000">OK.</span> <span m="4194109">Let</span> <span m="4194220">me</span> <span m="4194310">do</span> <span m="4194450">one</span> <span m="4194640">more.</span> <span m="4199660">Let's</span> <span m="4199710">do</span> <span m="4199970">that</span> <span m="4200090">over</span> <span m="4200360">here.</span> <span m="4205480">Just</span> <span m="4205790">to</span> <span m="4205870">make</span> <span m="4206080">sure</span> <span m="4206320">that</span> <span m="4206540">you really</span> <span m="4206880">understand</span> <span m="4207340">this</span> <span m="4207490">stuff,</span> <span m="4208870">I'm</span> <span m="4209010">going</span> <span m="4209250">to</span> <span m="4209410">count</span> <span m="4210190">hands</span> <span m="4210850">that</span> <span m="4211090">have</span> <span m="4213340">each</span> <span m="4213700">suit</span> <span m="4214420">in</span> <span m="4214620">it.</span> <span m="4215030">So</span> <span m="4216320">I</span> <span m="4216470">want</span> <span m="4216720">to</span> <span m="4216840">count</span> <span m="4217400">the</span> <span m="4217500">hands</span> <span m="4219060">with</span> <span m="4220055">every</span> <span m="4220985">suit.</span> </p>
<p><span m="4224710">So</span> <span m="4225700">as</span> <span m="4225890">an</span> <span m="4225960">example,</span> <span m="4234864">I</span> <span m="4235310">can</span> <span m="4235560">have,</span> <span m="4235760">say,</span> <span m="4236075">the 7</span> <span m="4236390">of</span> <span m="4236530">diamonds,</span> <span m="4238020">the</span> <span m="4238460">King</span> <span m="4238850">of</span> <span m="4239020">clubs,</span> <span m="4239920">the</span> <span m="4240030">3</span> <span m="4240620">of</span> <span m="4240770">diamonds,</span> <span m="4241825">the</span> <span m="4242200">Ace</span> <span m="4242560">of</span> <span m="4242840">hearts,</span> <span m="4244080">and</span> <span m="4244240">the</span> <span m="4244340">2</span> <span m="4244550">of</span> <span m="4244770">spades.</span> <span m="4246090">Why</span> <span m="4246660">is</span> <span m="4246820">this</span> <span m="4246980">a</span> <span m="4247030">proper</span> <span m="4247340">hand?</span> <span m="4247710">Because</span> <span m="4248390">I</span> <span m="4248480">can</span> <span m="4248580">see the</span> <span m="4248920">diamonds,</span> <span m="4249560">the</span> <span m="4249690">clubs,</span> <span m="4250910">the</span> <span m="4251000">hearts,</span> <span m="4251470">and</span> <span m="4251700">the</span> <span m="4251780">spades.</span> <span m="4252200">So</span> <span m="4252420">I</span> <span m="4252500">have</span> <span m="4254450">a hand</span> <span m="4254745">in</span> <span m="4255040">which</span> <span m="4255470">every</span> <span m="4255830">suit</span> <span m="4256270">is</span> <span m="4256430">represented.</span> </p>
<p><span m="4259370">So</span> <span m="4259660">what</span> <span m="4260190">do</span> <span m="4260310">I</span> <span m="4260420">do</span> <span m="4260570">for</span> <span m="4260710">my</span> <span m="4260820">representation?</span> <span m="4262520">Well,</span> <span m="4263450">I</span> <span m="4263610">can</span> <span m="4263820">have</span> <span m="4264830">the</span> <span m="4264930">values</span> <span m="4266790">of</span> <span m="4267050">each</span> <span m="4267390">suit</span> <span m="4275150">and</span> <span m="4275740">I</span> <span m="4275900">assign</span> <span m="4276180">a</span> <span m="4276230">specific</span> <span m="4276750">order</span> <span m="4277040">to</span> <span m="4277200">those.</span> <span m="4277870">So</span> <span m="4278570">I</span> <span m="4278680">want</span> <span m="4278850">to represent</span> <span m="4280090">for</span> <span m="4281020">the</span> <span m="4281130">diamonds</span> <span m="4281860">and</span> <span m="4282030">the</span> <span m="4282150">clubs</span> <span m="4283090">and</span> <span m="4283310">the</span> <span m="4283370">heart</span> <span m="4284260">and</span> <span m="4284440">the spade.</span> <span m="4285830">I</span> <span m="4285970">want</span> <span m="4286250">to</span> <span m="4286780">find</span> <span m="4287160">out</span> <span m="4287640">what</span> <span m="4287920">values</span> <span m="4288370">do</span> <span m="4288510">I</span> <span m="4288590">really</span> <span m="4288840">have.</span> <span m="4289440">So</span> <span m="4289940">values</span> <span m="4290330">of each</span> <span m="4290770">suit</span> <span m="4291690">in</span> <span m="4292140">the</span> <span m="4292320">order</span> <span m="4295950">D,</span> <span m="4296428">C,</span> <span m="4298570">hearts</span> <span m="4299200">and</span> <span m="4299590">spades.</span> </p>
<p><span m="4301320">And</span> <span m="4301530">how</span> <span m="4301670">many</span> <span m="4301960">ways</span> <span m="4302230">do I</span> <span m="4302400">have?</span> <span m="4303610">Well,</span> <span m="4303740">I</span> <span m="4303840">have</span> <span m="4304130">13</span> <span m="4304850">values</span> <span m="4305630">for</span> <span m="4306440">the</span> <span m="4306550">7,</span> <span m="4307876">I have</span> <span m="4308320">13</span> <span m="4308720">values</span> <span m="4309250">for</span> <span m="4309500">the</span> <span m="4309650">King--</span> <span m="4310280">for</span> <span m="4310670">this</span> <span m="4310880">one,</span> <span m="4311330">for</span> <span m="4311420">this</span> <span m="4311650">value,</span> <span m="4312370">13</span> <span m="4312800">values</span> <span m="4313150">for</span> <span m="4313300">that</span> <span m="4313450">one</span> <span m="4313700">and</span> <span m="4313760">13</span> <span m="4314210">values</span> <span m="4314580">for</span> <span m="4314690">this</span> <span m="4314920">one.</span> <span m="4316690">So</span> <span m="4316860">that</span> <span m="4317940">is</span> <span m="4318910">pretty</span> <span m="4319120">straightforward.</span> </p>
<p><span m="4320430">Now,</span> <span m="4320490">we</span> <span m="4320580">still</span> <span m="4320770">have</span> <span m="4320930">the</span> <span m="4321030">extra</span> <span m="4321410">card.</span> <span m="4322130">And</span> <span m="4322280">the</span> <span m="4322350">extra</span> <span m="4322740">card</span> <span m="4325210">has</span> <span m="4325490">a</span> <span m="4325600">suit</span> <span m="4329870">which</span> <span m="4330340">I</span> <span m="4330470">can</span> <span m="4330750">do</span> <span m="4331120">in</span> <span m="4331550">four</span> <span m="4331980">ways</span> <span m="4332510">because</span> <span m="4332770">of</span> <span m="4332890">four</span> <span m="4333140">choices.</span> <span m="4334950">And</span> <span m="4335080">finally,</span> <span m="4337380">it</span> <span m="4337540">also</span> <span m="4337780">has a</span> <span m="4338060">value.</span> <span m="4341500">And</span> <span m="4341650">the</span> <span m="4341810">value</span> <span m="4342310">is--</span> <span m="4344710">it's</span> <span m="4344950">actually--</span> <span m="4346210">so</span> <span m="4346470">how</span> <span m="4346780">much</span> <span m="4347030">is</span> <span m="4347170">this?</span> <span m="4347520">Can</span> <span m="4347870">this be</span> <span m="4348930">13?</span> <span m="4349430">Are there</span> <span m="4349820">13</span> <span m="4350190">values?</span> <span m="4351920">No,</span> <span m="4352060">not</span> <span m="4352300">really,</span> <span m="4352570">right?</span> <span m="4352910">Because</span> <span m="4353900">for</span> <span m="4354430">each</span> <span m="4355310">possible</span> <span m="4355880">given</span> <span m="4356040">suit</span> <span m="4356550">that</span> <span m="4356740">I've</span> <span m="4356970">chosen,</span> <span m="4357580">I</span> <span m="4357740">already</span> <span m="4358190">chose</span> <span m="4358590">a</span> <span m="4358730">value</span> <span m="4359230">before</span> <span m="4362820">when</span> <span m="4363880">I</span> <span m="4363930">chose</span> <span m="4365490">a</span> <span m="4365660">value</span> <span m="4365960">for</span> <span m="4366190">the</span> <span m="4366280">suit</span> <span m="4367320">that</span> <span m="4367660">I</span> <span m="4367950">did</span> <span m="4368080">over</span> <span m="4368320">here.</span> </p>
<p><span m="4368610">So</span> <span m="4368640">for</span> <span m="4368770">example,</span> <span m="4370070">this</span> <span m="4370340">one</span> <span m="4370590">cannot</span> <span m="4370960">be</span> <span m="4371120">the</span> <span m="4371210">7,</span> <span m="4372030">it</span> <span m="4372220">has</span> <span m="4372460">to</span> <span m="4372570">be</span> <span m="4372690">something</span> <span m="4373090">else.</span> <span m="4373330">There</span> <span m="4373420">are</span> <span m="4373530">12</span> <span m="4373870">choices.</span> <span m="4375100">If</span> <span m="4375820">I</span> <span m="4376130">would</span> <span m="4376300">have</span> <span m="4376430">chose a</span> <span m="4376810">different</span> <span m="4377140">suit,</span> <span m="4377550">for</span> <span m="4377710">example</span> <span m="4378210">hearts,</span> <span m="4378800">then</span> <span m="4378990">I</span> <span m="4379080">would</span> <span m="4379230">not</span> <span m="4379400">be</span> <span m="4379550">able</span> <span m="4379770">to</span> <span m="4379880">choose</span> <span m="4380140">the Ace</span> <span m="4380500">because</span> <span m="4380820">I already</span> <span m="4381120">have</span> <span m="4381310">chosen</span> <span m="4381800">the Ace</span> <span m="4382470">in</span> <span m="4382570">the</span> <span m="4382650">first</span> <span m="4382990">step.</span> <span m="4384050">So</span> <span m="4384120">I have</span> <span m="4384430">12</span> <span m="4386970">possibilities</span> <span m="4387660">here.</span> <span m="4387890">So</span> <span m="4388000">this</span> <span m="4388170">is</span> <span m="4388330">not</span> <span m="4388760">13.</span> <span m="4391260">You</span> <span m="4391420">may</span> <span m="4391590">want</span> <span m="4391700">to</span> <span m="4391800">check</span> <span m="4392070">that</span> <span m="4392220">again.</span> </p>
<p><span m="4393630">So</span> <span m="4395550">now</span> <span m="4395780">let's</span> <span m="4398050">give</span> <span m="4398210">the</span> <span m="4398290">representation</span> <span m="4399020">for this</span> <span m="4399440">one.</span> <span m="4399690">So</span> <span m="4399860">I</span> <span m="4399940">can</span> <span m="4400200">have</span> <span m="4401000">diamonds,</span> <span m="4401780">clubs--</span> <span m="4402110">so</span> <span m="4402260">let's</span> <span m="4402530">see,</span> <span m="4402690">in</span> <span m="4402770">the</span> <span m="4402860">order</span> <span m="4403180">of</span> <span m="4403380">diamonds,</span> <span m="4404030">clubs.</span> <span m="4404260">So</span> <span m="4404380">I</span> <span m="4404480">have</span> <span m="4404740">the</span> <span m="4404840">7,</span> <span m="4406130">I</span> <span m="4406370">have</span> <span m="4406640">a</span> <span m="4406810">King,</span> <span m="4408330">I</span> <span m="4408640">have</span> <span m="4408990">for</span> <span m="4409250">hearts</span> <span m="4409670">I</span> <span m="4409730">have an</span> <span m="4410140">Ace,</span> <span m="4411110">for</span> <span m="4411620">spades</span> <span m="4412140">I</span> <span m="4412220">have</span> <span m="4412630">a 2.</span> <span m="4414000">Then</span> <span m="4414370">I</span> <span m="4414560">have</span> <span m="4414960">the</span> <span m="4415070">suit</span> <span m="4415330">for the</span> <span m="4415610">extra</span> <span m="4416040">card</span> <span m="4416385">which</span> <span m="4416730">is a</span> <span m="4416820">diamonds.</span> <span m="4418450">And</span> <span m="4418960">then</span> <span m="4419240">a</span> <span m="4419470">value</span> <span m="4420130">which</span> <span m="4420380">is</span> <span m="4420610">a</span> <span m="4420790">3.</span> </p>
<p><span m="4422990">Now</span> <span m="4423180">we</span> <span m="4423540">are</span> <span m="4423700">going</span> <span m="4423960">to</span> <span m="4424180">check</span> <span m="4424560">according</span> <span m="4424960">to our</span> <span m="4425160">guidelines</span> <span m="4427510">over</span> <span m="4427790">here.</span> <span m="4429080">We're going</span> <span m="4429510">to</span> <span m="4429650">check</span> <span m="4429930">the</span> <span m="4430020">number</span> <span m="4430340">of</span> <span m="4430500">elements</span> <span m="4431370">that</span> <span m="4431710">are--</span> <span m="4433830">the</span> <span m="4433930">number</span> <span m="4434190">of</span> <span m="4434270">sequences</span> <span m="4434990">that</span> <span m="4435160">map</span> <span m="4435460">to</span> <span m="4435610">the</span> <span m="4435740">same</span> <span m="4438160">hand</span> <span m="4438590">over</span> <span m="4438720">here.</span> <span m="4439940">So</span> <span m="4440100">is</span> <span m="4440280">this a</span> <span m="4440410">bijection?</span> <span m="4442830">It's</span> <span m="4443020">not,</span> <span m="4443230">right?</span> </p>
<p><span m="4447100">So</span> <span m="4447800">why</span> <span m="4447970">is</span> <span m="4448100">this</span> <span m="4448290">not</span> <span m="4448500">a</span> <span m="4448700">bijection?</span> <span m="4449200">Well,</span> <span m="4449830">we</span> <span m="4449970">can</span> <span m="4450200">also</span> <span m="4450750">have</span> <span m="4451630">the</span> <span m="4451950">following</span> <span m="4453730">situation</span> <span m="4454450">where</span> <span m="4454530">we</span> <span m="4454970">swap</span> <span m="4455500">the</span> <span m="4455610">3</span> <span m="4456040">diamonds</span> <span m="4456120">and the</span> <span m="4456340">7</span> <span m="4456800">diamonds</span> <span m="4457250">around.</span> <span m="4458340">So</span> <span m="4458570">we</span> <span m="4458690">choose</span> <span m="4459330">the</span> <span m="4459480">3</span> <span m="4459940">for</span> <span m="4460360">the</span> <span m="4460470">diamonds</span> <span m="4461800">and</span> <span m="4462010">then</span> <span m="4462170">we</span> <span m="4462280">have</span> <span m="4462560">the</span> <span m="4462680">King,</span> <span m="4463260">the</span> <span m="4463470">Ace</span> <span m="4464010">and</span> <span m="4464200">the</span> <span m="4464370">2.</span> <span m="4465440">then</span> <span m="4465580">we</span> <span m="4465660">have</span> <span m="4465880">the</span> <span m="4465990">fifth</span> <span m="4466380">card</span> <span m="4466770">which</span> <span m="4466950">is</span> <span m="4467080">also</span> <span m="4467540">diamonds,</span> <span m="4468070">but</span> <span m="4468250">now</span> <span m="4468460">it's</span> <span m="4468660">the</span> <span m="4468710">7.</span> </p>
<p><span m="4471530">Well,</span> <span m="4471660">this</span> <span m="4471920">one</span> <span m="4472620">also</span> <span m="4472990">maps</span> <span m="4473680">to</span> <span m="4473840">this</span> <span m="4476710">particular</span> <span m="4477230">hand.</span> <span m="4477720">So</span> <span m="4477910">we</span> <span m="4478000">have</span> <span m="4478260">a 2-to-1</span> <span m="4478940">mapping.</span> <span m="4481370">And</span> <span m="4482080">we</span> <span m="4482200">conclude</span> <span m="4482800">that</span> <span m="4482890">the</span> <span m="4483030">total</span> <span m="4483970">number</span> <span m="4484610">of</span> <span m="4484680">hands,</span> <span m="4485420">where</span> <span m="4485650">every</span> <span m="4485860">suit</span> <span m="4486020">is</span> <span m="4486520">represented,</span> <span m="4487620">is</span> <span m="4488130">13</span> <span m="4488630">to</span> <span m="4488750">the</span> <span m="4488850">power</span> <span m="4489150">4</span> <span m="4490230">times</span> <span m="4490630">4</span> <span m="4491340">times</span> <span m="4491690">12</span> <span m="4492290">by the</span> <span m="4492430">generalized</span> <span m="4493420">product</span> <span m="4494030">rule.</span> <span m="4494600">And</span> <span m="4494750">now</span> <span m="4494880">by</span> <span m="4495040">the</span> <span m="4495130">division</span> <span m="4495650">rule,</span> <span m="4496450">we</span> <span m="4496570">have</span> <span m="4496770">to</span> <span m="4496880">still</span> <span m="4497170">divide</span> <span m="4497670">by</span> <span m="4497820">2.</span> <span m="4498970">So</span> <span m="4499140">that's</span> <span m="4499400">the total</span> <span m="4499888">number.</span> </p>
<p><span m="4503310">OK.</span> <span m="4503620">So</span> <span m="4503780">now</span> <span m="4503970">we</span> <span m="4504100">come</span> <span m="4504390">to</span> <span m="4506180">combinatorial</span> <span m="4506940">proofs.</span> <span m="4507950">So</span> <span m="4508150">this</span> <span m="4508220">is</span> <span m="4508450">a</span> <span m="4511102">new</span> <span m="4511480">proof</span> <span m="4511820">technique.</span> <span m="4517690">And</span> <span m="4517870">what</span> <span m="4518050">we're</span> <span m="4518140">going</span> <span m="4518380">to</span> <span m="4518470">do</span> <span m="4518630">is</span> <span m="4518760">we're</span> <span m="4518840">going</span> <span m="4519090">to</span> <span m="4519300">count</span> <span m="4519610">a</span> <span m="4519700">set</span> <span m="4520070">in</span> <span m="4520230">two</span> <span m="4520490">different</span> <span m="4520900">ways.</span> <span m="4522610">And</span> <span m="4523340">that</span> <span m="4523550">will</span> <span m="4523610">lead</span> <span m="4523890">to</span> <span m="4524210">a</span> <span m="4525170">combinatorial</span> <span m="4525600">equation.</span> </p>
<p><span m="4527440">So</span> <span m="4529590">the</span> <span m="4529660">whole</span> <span m="4529900">idea</span> <span m="4530300">is</span> <span m="4532420">as</span> <span m="4532510">follows.</span> <span m="4532890">So</span> <span m="4533270">let</span> <span m="4533420">me</span> <span m="4533510">give</span> <span m="4533690">an</span> <span m="4533760">example</span> <span m="4534210">first.</span> <span m="4539640">So</span> <span m="4539840">for</span> <span m="4540080">a</span> <span m="4540440">combinatorial</span> <span m="4541070">proof,</span> <span m="4542730">for</span> <span m="4542950">example,</span> <span m="4544270">suppose</span> <span m="4544970">I</span> <span m="4545200">have</span> <span m="4547390">n</span> <span m="4547810">shirts</span> <span m="4550920">and</span> <span m="4551160">I</span> <span m="4551200">want</span> <span m="4551460">to</span> <span m="4551720">choose--</span> <span m="4552880">I</span> <span m="4553010">want</span> <span m="4553200">to</span> <span m="4553340">keep</span> <span m="4554490">k</span> <span m="4555390">and</span> <span m="4555610">I</span> <span m="4555640">want</span> <span m="4555880">to</span> <span m="4556070">trash</span> <span m="4559180">n</span> <span m="4559350">minus</span> <span m="4559760">k.</span> </p>
<p><span m="4561760">So</span> <span m="4562470">how</span> <span m="4562600">do</span> <span m="4562780">I</span> <span m="4562920">count</span> <span m="4565340">the</span> <span m="4565450">number</span> <span m="4565680">of</span> <span m="4565780">choices</span> <span m="4566200">that</span> <span m="4566440">I</span> <span m="4566520">have?</span> <span m="4567050">Well,</span> <span m="4567380">I</span> <span m="4567490">can</span> <span m="4567720">choose,</span> <span m="4568150">out</span> <span m="4568430">of</span> <span m="4568640">n,</span> <span m="4569440">I</span> <span m="4569580">can</span> <span m="4569750">choose</span> <span m="4570010">the</span> <span m="4570110">k</span> <span m="4570513">keepers.</span> <span m="4573490">But</span> <span m="4573620">I also</span> <span m="4573930">can</span> <span m="4574110">count it</span> <span m="4574350">in</span> <span m="4574460">a</span> <span m="4574560">different</span> <span m="4575020">way.</span> <span m="4575460">I</span> <span m="4575570">can</span> <span m="4575750">also</span> <span m="4575930">count</span> <span m="4576290">this</span> <span m="4577070">as</span> <span m="4578820">how</span> <span m="4579030">many</span> <span m="4579320">ways</span> <span m="4579680">I</span> <span m="4579820">can</span> <span m="4580780">select</span> <span m="4581340">the</span> <span m="4581460">trashers.</span> <span m="4582390">So</span> <span m="4582960">there</span> <span m="4583130">are</span> <span m="4583210">n</span> <span m="4583540">minus</span> <span m="4584010">k</span> <span m="4585260">out</span> <span m="4585420">of</span> <span m="4585610">n</span> <span m="4586680">shirts</span> <span m="4587160">that I</span> <span m="4587330">will</span> <span m="4587480">trash.</span> </p>
<p><span m="4588910">So</span> <span m="4589080">there</span> <span m="4589180">are</span> <span m="4589260">two</span> <span m="4589470">different</span> <span m="4589890">ways</span> <span m="4591260">to</span> <span m="4591330">count</span> <span m="4591720">the</span> <span m="4591820">same</span> <span m="4594760">choices</span> <span m="4595220">that</span> <span m="4595490">I</span> <span m="4595560">have.</span> <span m="4596670">So</span> <span m="4598500">we</span> <span m="4598630">have</span> <span m="4598920">an</span> <span m="4599340">equation</span> <span m="4599940">over</span> <span m="4600230">here.</span> <span m="4602130">And</span> <span m="4602310">you</span> <span m="4602410">already</span> <span m="4602680">know</span> <span m="4602870">this</span> <span m="4603100">one</span> <span m="4603270">because</span> <span m="4604130">by</span> <span m="4604260">the</span> <span m="4604360">definition</span> <span m="4605040">of</span> <span m="4605260">the</span> <span m="4605650">binomial</span> <span m="4606160">coefficient,</span> <span m="4607880">we</span> <span m="4608050">have</span> <span m="4608540">that this</span> <span m="4608850">is</span> <span m="4608960">equal</span> <span m="4609290">to</span> <span m="4609430">this.</span> <span m="4611080">And</span> <span m="4611280">so</span> <span m="4611510">it's</span> <span m="4612070">pretty</span> <span m="4612350">straightforward.</span> <span m="4613000">It's</span> <span m="4613200">equal</span> <span m="4613410">to</span> <span m="4613510">this</span> <span m="4613740">one</span> <span m="4613910">over</span> <span m="4614170">here.</span> </p>
<p><span m="4616070">But</span> <span m="4616160">the</span> <span m="4616300">idea</span> <span m="4616670">is</span> <span m="4616770">that we're</span> <span m="4616890">going</span> <span m="4617280">to</span> <span m="4617430">count</span> <span m="4618990">a</span> <span m="4619070">set</span> <span m="4619440">in</span> <span m="4619650">two</span> <span m="4619840">different</span> <span m="4620230">ways.</span> <span m="4620510">So</span> <span m="4620610">another</span> <span m="4622080">example</span> <span m="4622980">is</span> <span m="4625480">where</span> <span m="4625760">we</span> <span m="4625910">have</span> <span m="4628090">the</span> <span m="4628240">following.</span> <span m="4628820">We</span> <span m="4628870">choose</span> <span m="4630040">a</span> <span m="4630260">team</span> <span m="4633280">of</span> <span m="4634138">k</span> <span m="4638290">elements.</span> <span m="4646790">Of k</span> <span m="4646980">elements</span> <span m="4647420">out</span> <span m="4647800">of--</span> <span m="4650470">oh,</span> <span m="4650740">actually,</span> <span m="4651090">I'm</span> <span m="4651180">talking</span> <span m="4651460">about</span> <span m="4651680">a</span> <span m="4651810">team.</span> <span m="4652280">I</span> <span m="4652400">was</span> <span m="4652830">talking</span> <span m="4653210">about</span> <span m="4653390">students</span> <span m="4653910">here.</span> <span m="4654640">Of</span> <span m="4655060">k</span> <span m="4655280">students.</span> </p>
<p><span m="4658620">Out</span> <span m="4659020">of n</span> <span m="4659230">students.</span> <span m="4660750">Now,</span> <span m="4660890">how</span> <span m="4661030">many</span> <span m="4661270">ways</span> <span m="4661540">can</span> <span m="4661700">I</span> <span m="4661900">do</span> <span m="4661970">this?</span> <span m="4664260">Well,</span> <span m="4665410">n</span> <span m="4665590">choose</span> <span m="4665890">k,</span> <span m="4666130">right?</span> <span m="4667450">Well,</span> <span m="4667560">there is</span> <span m="4667790">a way</span> <span m="4668040">to</span> <span m="4668210">count</span> <span m="4668470">this</span> <span m="4668590">differently.</span> <span m="4669440">I</span> <span m="4669560">could</span> <span m="4669760">also</span> <span m="4670040">say,</span> <span m="4672730">well,</span> <span m="4673050">let's</span> <span m="4673360">first</span> <span m="4673880">count</span> <span m="4675590">the</span> <span m="4675650">total</span> <span m="4676030">number</span> <span m="4676360">of</span> <span m="4676530">ways</span> <span m="4677420">in</span> <span m="4677660">which</span> <span m="4678060">I</span> <span m="4678200">choose</span> <span m="4679850">k</span> <span m="4680260">students</span> <span m="4681050">that</span> <span m="4681210">includes</span> <span m="4681900">Bob.</span> <span m="4682980">So</span> <span m="4683180">let's</span> <span m="4683420">do</span> <span m="4683580">that.</span> </p>
<p><span m="4685260">So</span> <span m="4685350">Bob</span> <span m="4685610">is</span> <span m="4685740">one</span> <span m="4685920">of</span> <span m="4686050">the</span> <span m="4686130">students,</span> <span m="4687580">and</span> <span m="4692780">if</span> <span m="4693000">I</span> <span m="4693050">count</span> <span m="4693550">the</span> <span m="4693640">number</span> <span m="4693940">of</span> <span m="4694160">teams</span> <span m="4697020">with</span> <span m="4697430">Bob,</span> <span m="4698860">how</span> <span m="4699020">many</span> <span m="4699270">choices</span> <span m="4699730">do</span> <span m="4699880">I</span> <span m="4699990">have?</span> <span m="4704480">I</span> <span m="4704610">choose</span> <span m="4704960">Bob.</span> <span m="4705590">I</span> <span m="4705700">still</span> <span m="4705970">need</span> <span m="4706130">to</span> <span m="4706230">choose</span> <span m="4706600">k</span> <span m="4706840">minus</span> <span m="4707240">1</span> <span m="4707460">students</span> <span m="4708620">out</span> <span m="4708820">of</span> <span m="4708990">the</span> <span m="4709080">remaining</span> <span m="4709650">n</span> <span m="4709870">minus</span> <span m="4710200">1</span> <span m="4710320">students.</span> <span m="4710770">So I</span> <span m="4710890">have</span> <span m="4711150">taken</span> <span m="4711500">out</span> <span m="4711690">Bob.</span> <span m="4712330">So there</span> <span m="4712620">are n</span> <span m="4712830">minus</span> <span m="4713210">1</span> <span m="4713370">students</span> <span m="4713870">left</span> <span m="4714700">and</span> <span m="4714840">I</span> <span m="4714890">still</span> <span m="4715240">need</span> <span m="4715410">to</span> <span m="4715510">choose</span> <span m="4715860">k</span> <span m="4716050">minus</span> <span m="4716420">1</span> <span m="4716590">students.</span> </p>
<p><span m="4719920">The</span> <span m="4720140">other</span> <span m="4720790">possibility</span> <span m="4721530">is</span> <span m="4721750">that</span> <span m="4721880">we</span> <span m="4721990">have</span> <span m="4722220">teams</span> <span m="4723230">that--</span> <span m="4725450">in</span> <span m="4725590">which</span> <span m="4725820">Bob is</span> <span m="4726220">not</span> <span m="4726370">represented.</span> <span m="4728880">So</span> <span m="4729610">now</span> <span m="4729920">we</span> <span m="4730030">need</span> <span m="4730240">to</span> <span m="4730350">choose</span> <span m="4731440">k</span> <span m="4732080">students</span> <span m="4734060">out</span> <span m="4734310">off</span> <span m="4735030">all</span> <span m="4735220">the</span> <span m="4735310">students</span> <span m="4736040">minus</span> <span m="4736490">Bob.</span> <span m="4736980">So</span> <span m="4737270">we</span> <span m="4737360">have</span> <span m="4738360">n</span> <span m="4738540">minus</span> <span m="4738920">1</span> <span m="4739180">choose</span> <span m="4739560">k.</span> <span m="4741190">So</span> <span m="4741410">by</span> <span m="4741570">the sum</span> <span m="4742070">rule,</span> <span m="4743080">we</span> <span m="4743130">can</span> <span m="4743280">just</span> <span m="4743660">add</span> <span m="4743940">those</span> <span m="4744170">two</span> <span m="4744340">together</span> <span m="4745350">and</span> <span m="4745410">that</span> <span m="4745540">should</span> <span m="4745710">be</span> <span m="4745990">equal</span> <span m="4747120">to</span> <span m="4748060">all</span> <span m="4748280">the</span> <span m="4748380">possibilities</span> <span m="4749050">to</span> <span m="4749160">choose</span> <span m="4749560">team of</span> <span m="4749980">k</span> <span m="4750180">students.</span> </p>
<p><span m="4752070">So</span> <span m="4752290">what</span> <span m="4752420">do</span> <span m="4752520">I</span> <span m="4752610">see?</span> <span m="4752990">I</span> <span m="4753060">see that</span> <span m="4753460">n</span> <span m="4753660">minus</span> <span m="4754080">1</span> <span m="4754360">choose</span> <span m="4754860">k</span> <span m="4755060">minus</span> <span m="4755500">1</span> <span m="4756210">plus</span> <span m="4756980">n</span> <span m="4757210">minus</span> <span m="4757590">1</span> <span m="4757830">choose</span> <span m="4758070">k</span> <span m="4759650">should</span> <span m="4759880">be</span> <span m="4760070">equal</span> <span m="4760610">to--</span> <span m="4761570">well,</span> <span m="4762540">before</span> <span m="4762920">I</span> <span m="4762960">counted it</span> <span m="4763270">as</span> <span m="4763750">this--</span> <span m="4764030">so</span> <span m="4764080">it</span> <span m="4764140">should</span> <span m="4764300">be</span> <span m="4764490">equal</span> <span m="4765650">to</span> <span m="4768090">n</span> <span m="4768320">choose</span> <span m="4768670">k.</span> <span m="4769330">And</span> <span m="4769470">this</span> <span m="4769640">is</span> <span m="4769800">called</span> <span m="4770130">Pascal's</span> <span m="4772020">Identity.</span> <span m="4775820">And</span> <span m="4775980">the</span> <span m="4776050">general</span> <span m="4776850">idea</span> <span m="4777410">is is</span> <span m="4777670">that</span> <span m="4777930">we</span> <span m="4778040">do</span> <span m="4778160">the</span> <span m="4778270">following.</span> <span m="4779110">So</span> <span m="4779450">we</span> <span m="4779630">are</span> <span m="4782910">actually</span> <span m="4784650">counting</span> <span m="4785280">a</span> <span m="4785400">set</span> <span m="4785720">in</span> <span m="4785870">two</span> <span m="4786060">different</span> <span m="4786450">ways.</span> <span m="4786710">That's</span> <span m="4786880">what</span> <span m="4787030">we</span> <span m="4787130">did</span> <span m="4787340">so</span> <span m="4787570">far.</span> </p>
<p><span m="4789990">And</span> <span m="4790130">when</span> <span m="4790260">you</span> <span m="4790370">solve</span> <span m="4790720">these</span> <span m="4790910">types</span> <span m="4791190">of</span> <span m="4791290">problems,</span> <span m="4792980">the</span> <span m="4793090">difficulty</span> <span m="4793830">is</span> <span m="4794220">that</span> <span m="4794400">you</span> <span m="4794520">will</span> <span m="4794640">need</span> <span m="4794880">to</span> <span m="4795110">define</span> <span m="4797820">the</span> <span m="4797950">set</span> <span m="4798280">S.</span> <span m="4798740">So</span> <span m="4798910">usually</span> <span m="4799380">you</span> <span m="4799480">get</span> <span m="4799680">a</span> <span m="4799750">problem,</span> <span m="4800130">you</span> <span m="4800240">need</span> <span m="4800400">to</span> <span m="4801450">prove</span> <span m="4801790">some</span> <span m="4802150">equation</span> <span m="4802770">like</span> <span m="4803000">this.</span> <span m="4803830">So</span> <span m="4803890">you</span> <span m="4804030">define</span> <span m="4804560">a</span> <span m="4804610">set</span> <span m="4804910">S.</span> <span m="4805270">That's</span> <span m="4805570">the</span> <span m="4805650">hard</span> <span m="4805980">part</span> <span m="4806300">of</span> <span m="4806410">it.</span> </p>
<p><span m="4807790">Then</span> <span m="4808490">we</span> <span m="4808650">are</span> <span m="4808680">going</span> <span m="4808890">to</span> <span m="4809060">show</span> <span m="4810900">that</span> <span m="4811200">the</span> <span m="4811300">cardinality</span> <span m="4812040">of</span> <span m="4812230">S</span> <span m="4812590">is</span> <span m="4812720">some</span> <span m="4813330">number</span> <span m="4813820">n,</span> <span m="4814170">say,</span> <span m="4815230">in</span> <span m="4815440">one</span> <span m="4815680">way.</span> <span m="4817550">So</span> <span m="4818190">we</span> <span m="4818300">have</span> <span m="4818480">one</span> <span m="4818700">method</span> <span m="4819050">to</span> <span m="4819170">do</span> <span m="4819350">this.</span> <span m="4819770">By</span> <span m="4820225">counting.</span> <span m="4823870">And</span> <span m="4826100">we</span> <span m="4826300">will</span> <span m="4826430">show</span> <span m="4828020">that</span> <span m="4828300">S</span> <span m="4828850">is</span> <span m="4829180">also</span> <span m="4829610">equal</span> <span m="4830370">to</span> <span m="4831390">some</span> <span m="4831640">other</span> <span m="4831870">number--</span> <span m="4833210">like,</span> <span m="4833430">in</span> <span m="4833500">this</span> <span m="4833720">case,</span> <span m="4834170">we</span> <span m="4834310">had</span> <span m="4834570">one</span> <span m="4834850">number</span> <span m="4835260">and</span> <span m="4835380">this</span> <span m="4835550">was</span> <span m="4835750">my</span> <span m="4835850">second</span> <span m="4836260">one--</span> <span m="4837520">also</span> <span m="4837810">by</span> <span m="4837940">counting.</span> <span m="4838600">And</span> <span m="4838830">then</span> <span m="4838920">we</span> <span m="4839020">can</span> <span m="4839200">conclude</span> <span m="4840960">that</span> <span m="4842580">n--</span> <span m="4843270">so</span> <span m="4843440">we</span> <span m="4843550">conclude</span> <span m="4845628">that</span> <span m="4846104">n</span> <span m="4846580">equals</span> <span m="4847056">m.</span> </p>
<p><span m="4850400">So</span> <span m="4850840">we're</span> <span m="4850980">almost</span> <span m="4851410">done</span> <span m="4851660">because</span> <span m="4852050">I'm</span> <span m="4852140">going</span> <span m="4852440">to</span> <span m="4852580">prove</span> <span m="4852890">to</span> <span m="4853020">you</span> <span m="4853360">a</span> <span m="4853470">very</span> <span m="4853720">simple</span> <span m="4854680">equation</span> <span m="4856480">using</span> <span m="4856890">this</span> <span m="4857700">technique.</span> <span m="4858980">But</span> <span m="4859220">it is</span> <span m="4860770">not</span> <span m="4860940">as</span> <span m="4861080">trivial</span> <span m="4861670">as</span> <span m="4861810">the</span> <span m="4861920">one</span> <span m="4862110">that we</span> <span m="4862320">saw</span> <span m="4862816">just now.</span> <span m="4866790">What</span> <span m="4866930">we</span> <span m="4867070">do</span> <span m="4867340">now</span> <span m="4867810">is</span> <span m="4868080">the</span> <span m="4868190">following.</span> <span m="4868570">We want</span> <span m="4868730">to</span> <span m="4869010">prove</span> <span m="4869380">a</span> <span m="4869450">theorem</span> <span m="4870270">that</span> <span m="4870420">says</span> <span m="4871380">that</span> <span m="4871510">if</span> <span m="4871660">a</span> <span m="4872000">sum</span> <span m="4873140">r</span> <span m="4873800">is</span> <span m="4873990">0</span> <span m="4874410">all</span> <span m="4874600">the</span> <span m="4874700">way</span> <span m="4874910">up</span> <span m="4875070">to</span> <span m="4875270">n,</span> <span m="4876800">then,</span> <span m="4877320">if</span> <span m="4877510">you</span> <span m="4877610">take</span> <span m="4877830">the</span> <span m="4877930">product</span> <span m="4878360">of</span> <span m="4878480">the</span> <span m="4878570">following</span> <span m="4879020">binomial</span> <span m="4879710">coefficients</span> <span m="4880520">n</span> <span m="4880640">choose</span> <span m="4880970">r</span> <span m="4882070">times</span> <span m="4882360">2n</span> <span m="4883030">choose</span> <span m="4884210">n</span> <span m="4884460">minus</span> <span m="4884880">r,</span> <span m="4886370">this</span> <span m="4886610">is</span> <span m="4886730">equal</span> <span m="4887260">to</span> <span m="4887410">3n</span> <span m="4888880">choose</span> <span m="4889330">n.</span> <span m="4890460">We</span> <span m="4890570">need</span> <span m="4890730">to</span> <span m="4890840">prove</span> <span m="4891180">this.</span> </p>
<p><span m="4892670">So</span> <span m="4892830">we</span> <span m="4892910">need</span> <span m="4893100">to</span> <span m="4893530">find</span> <span m="4893840">the</span> <span m="4893950">proper</span> <span m="4894280">set.</span> <span m="4894950">So</span> <span m="4895130">how</span> <span m="4895330">can</span> <span m="4895440">we</span> <span m="4895590">think</span> <span m="4895840">about</span> <span m="4896210">it?</span> <span m="4896410">Well,</span> <span m="4896790">the</span> <span m="4896950">idea</span> <span m="4897140">is</span> <span m="4897330">that we're</span> <span m="4897430">probably</span> <span m="4897790">going</span> <span m="4898000">to</span> <span m="4898100">choose</span> <span m="4898510">n</span> <span m="4898800">elements</span> <span m="4899330">out</span> <span m="4899570">of</span> <span m="4901460">3n</span> <span m="4902010">elements.</span> <span m="4903020">We're going</span> <span m="4903390">to do it</span> <span m="4903620">in</span> <span m="4903720">a</span> <span m="4903780">special</span> <span m="4904310">way.</span> <span m="4905620">And</span> <span m="4905920">so</span> <span m="4908210">the</span> <span m="4908320">proof</span> <span m="4908680">will</span> <span m="4908750">be</span> <span m="4908940">as</span> <span m="4909080">follows.</span> </p>
<p><span m="4910890">We</span> <span m="4911110">take</span> <span m="4911410">S</span> <span m="4911750">to</span> <span m="4911880">be</span> <span m="4913270">all</span> <span m="4913520">the</span> <span m="4913620">subsets</span> <span m="4916980">of</span> <span m="4918280">n</span> <span m="4918705">balls</span> <span m="4922570">chosen</span> <span m="4925050">from</span> <span m="4925310">a</span> <span m="4925370">basket</span> <span m="4929680">of</span> <span m="4931400">n</span> <span m="4931900">red</span> <span m="4932280">balls</span> <span m="4935220">and</span> <span m="4936660">2n</span> <span m="4938030">green</span> <span m="4938390">balls.</span> <span m="4939990">And</span> <span m="4940280">this</span> <span m="4940460">will</span> <span m="4940530">do</span> <span m="4940640">the</span> <span m="4940770">trick.</span> <span m="4942540">So</span> <span m="4943320">how</span> <span m="4943480">do</span> <span m="4943590">we</span> <span m="4943700">do</span> <span m="4943880">this?</span> </p>
<p><span m="4945310">We're</span> <span m="4945430">going</span> <span m="4945750">to</span> <span m="4949030">first</span> <span m="4949450">choose</span> <span m="4950150">the</span> <span m="4950310">red</span> <span m="4950570">balls</span> <span m="4952180">and</span> <span m="4952380">then</span> <span m="4952690">the</span> <span m="4952800">green</span> <span m="4953080">balls.</span> <span m="4955110">So</span> <span m="4955310">let's</span> <span m="4955570">first</span> <span m="4957100">count</span> <span m="4957460">S</span> <span m="4957715">in</span> <span m="4957970">the</span> <span m="4958070">very</span> <span m="4958340">easy</span> <span m="4958740">way.</span> <span m="4960360">Well,</span> <span m="4961090">I</span> <span m="4961230">just</span> <span m="4961640">choose</span> <span m="4961990">n</span> <span m="4962300">balls</span> <span m="4962800">out</span> <span m="4962920">of a</span> <span m="4963140">basket</span> <span m="4963620">of</span> <span m="4963955">3n</span> <span m="4964290">balls.</span> <span m="4964570">I don't</span> <span m="4964870">care</span> <span m="4965220">how</span> <span m="4965370">many</span> <span m="4965610">green</span> <span m="4965990">or red</span> <span m="4966310">balls</span> <span m="4966630">I have.</span> <span m="4967100">So</span> <span m="4967280">I</span> <span m="4967380">have</span> <span m="4967550">really</span> <span m="4969454">3n</span> <span m="4970460">choose</span> <span m="4970850">n</span> <span m="4971120">choices.</span> <span m="4973990">This</span> <span m="4974140">is</span> <span m="4974280">the</span> <span m="4974370">number</span> <span m="4975850">of</span> <span m="4977666">n</span> <span m="4978120">element</span> <span m="4979226">subsets</span> <span m="4983330">from</span> <span m="4984550">a</span> <span m="4984680">3n</span> <span m="4987040">element</span> <span m="4987940">set.</span> <span m="4989660">So</span> <span m="4989860">that's</span> <span m="4990040">easy.</span> </p>
<p><span m="4994710">Now,</span> <span m="4994860">we</span> <span m="4994960">can</span> <span m="4995100">also</span> <span m="4995430">count</span> <span m="4995700">it in</span> <span m="4995780">a</span> <span m="4995990">different</span> <span m="4996340">way.</span> <span m="4996860">We</span> <span m="4996980">first</span> <span m="4997330">of</span> <span m="4997400">all</span> <span m="4998370">wonder</span> <span m="4998870">about</span> <span m="4999270">the</span> <span m="4999440">number</span> <span m="4999800">of</span> <span m="4999950">subsets</span> <span m="5002480">with</span> <span m="5003000">exactly</span> <span m="5004150">r</span> <span m="5004790">red</span> <span m="5005270">balls.</span> <span m="5006430">So</span> <span m="5006640">how</span> <span m="5006770">many</span> <span m="5007080">are</span> <span m="5007230">there?</span> <span m="5008710">Well,</span> <span m="5008860">we</span> <span m="5008980">choose</span> <span m="5009560">r</span> <span m="5010170">red</span> <span m="5010530">balls</span> <span m="5010900">so we</span> <span m="5010990">have</span> <span m="5011290">n</span> <span m="5011680">red</span> <span m="5011980">balls.</span> <span m="5012220">So</span> <span m="5012350">we</span> <span m="5012430">need</span> <span m="5012590">to</span> <span m="5012700">choose</span> <span m="5013150">r</span> <span m="5013460">out</span> <span m="5013700">of</span> <span m="5013840">those.</span> <span m="5015760">Now,</span> <span m="5015950">in</span> <span m="5016180">total,</span> <span m="5016930">I</span> <span m="5017090">need</span> <span m="5017320">to</span> <span m="5017480">choose</span> <span m="5017810">n</span> <span m="5018130">balls.</span> <span m="5018690">So</span> <span m="5018860">I</span> <span m="5018920">still</span> <span m="5019600">need</span> <span m="5019810">to</span> <span m="5019910">choose</span> <span m="5020210">n</span> <span m="5020370">minus</span> <span m="5020800">r</span> <span m="5021050">balls</span> <span m="5021500">out</span> <span m="5021700">of</span> <span m="5021850">the set</span> <span m="5022110">of</span> <span m="5022370">green</span> <span m="5022670">balls.</span> </p>
<p><span m="5023830">I have</span> <span m="5023990">2n</span> <span m="5024500">green</span> <span m="5024790">balls.</span> <span m="5026950">I'm</span> <span m="5027090">going</span> <span m="5027380">to</span> <span m="5027510">choose</span> <span m="5028120">n</span> <span m="5028490">minus</span> <span m="5028970">r</span> <span m="5029590">green</span> <span m="5029910">balls</span> <span m="5030350">over</span> <span m="5030470">here.</span> <span m="5030630">So</span> <span m="5030830">these</span> <span m="5030970">are</span> <span m="5031220">the</span> <span m="5031300">red</span> <span m="5031550">balls</span> <span m="5031930">that</span> <span m="5032250">I</span> <span m="5034200">select</span> <span m="5034740">and</span> <span m="5034900">here</span> <span m="5035130">are</span> <span m="5035280">the</span> <span m="5035340">green</span> <span m="5035620">balls.</span> <span m="5037420">Now</span> <span m="5037650">we'll</span> <span m="5037720">use</span> <span m="5037980">the</span> <span m="5038040">sum</span> <span m="5038330">rule.</span> <span m="5039500">And</span> <span m="5039690">I</span> <span m="5039840">add</span> <span m="5040080">all</span> <span m="5040340">the</span> <span m="5040420">different</span> <span m="5040890">possible</span> <span m="5041120">subsets</span> <span m="5041760">with</span> <span m="5041890">0</span> <span m="5042350">red</span> <span m="5042590">balls,</span> <span m="5042880">with</span> <span m="5043040">1</span> <span m="5043260">red</span> <span m="5043540">ball,</span> <span m="5043830">2</span> <span m="5044080">red</span> <span m="5044320">balls,</span> <span m="5045310">all</span> <span m="5045510">the</span> <span m="5045580">way</span> <span m="5045780">up</span> <span m="5045850">to</span> <span m="5046010">n</span> <span m="5046240">red</span> <span m="5046480">balls.</span> <span m="5047960">So essentially,</span> <span m="5048320">I</span> <span m="5048360">get</span> <span m="5048610">the</span> <span m="5048680">sum</span> <span m="5052820">by</span> <span m="5052980">the sum</span> <span m="5053450">rule</span> <span m="5056390">of</span> <span m="5058050">r</span> <span m="5058840">0</span> <span m="5059360">all</span> <span m="5059570">the</span> <span m="5059660">way</span> <span m="5059860">up</span> <span m="5059970">to</span> <span m="5060150">n</span> <span m="5061970">where</span> <span m="5062670">I count</span> <span m="5064150">the</span> <span m="5064250">number</span> <span m="5064660">of</span> <span m="5064770">subsets</span> <span m="5065330">with</span> <span m="5065640">exactly</span> <span m="5066280">r</span> <span m="5066570">red</span> <span m="5066830">balls.</span> <span m="5067730">So</span> <span m="5067970">that</span> <span m="5068100">means</span> <span m="5068340">I</span> <span m="5068380">can</span> <span m="5068580">choose</span> <span m="5069010">r</span> <span m="5069350">out</span> <span m="5069580">of</span> <span m="5069860">the n</span> <span m="5070070">red</span> <span m="5070310">balls</span> <span m="5071110">and</span> <span m="5071270">another</span> <span m="5072090">n</span> <span m="5072310">minus</span> <span m="5072700">r</span> <span m="5073050">out of</span> <span m="5073370">the</span> <span m="5073510">2n</span> <span m="5073800">balls--</span> <span m="5074690">2n</span> <span m="5075270">green</span> <span m="5075530">balls.</span> </p>
<p><span m="5077190">So</span> <span m="5077390">now</span> <span m="5077580">we</span> <span m="5077720">equate</span> <span m="5078120">those</span> <span m="5078380">two</span> <span m="5078530">together</span> <span m="5079360">and</span> <span m="5079490">that</span> <span m="5079640">proves</span> <span m="5079930">the</span> <span m="5079980">theorem.</span> <span m="5081430">So</span> <span m="5082060">the</span> <span m="5082190">really</span> <span m="5082500">hard</span> <span m="5082930">part</span> <span m="5083370">in</span> <span m="5083500">this</span> <span m="5084060">type</span> <span m="5084620">of</span> <span m="5084750">proofs</span> <span m="5085710">is</span> <span m="5086450">that</span> <span m="5086600">you</span> <span m="5086710">need</span> <span m="5086960">some</span> <span m="5087130">creativity</span> <span m="5088060">to</span> <span m="5088270">find</span> <span m="5088800">and</span> <span m="5088940">define</span> <span m="5089630">that</span> <span m="5089810">set</span> <span m="5090100">S</span> <span m="5091130">and</span> <span m="5091330">see</span> <span m="5091560">how</span> <span m="5091750">that</span> <span m="5091950">would</span> <span m="5092120">work.</span> <span m="5093020">And</span> <span m="5093170">the</span> <span m="5093230">way</span> <span m="5093360">to</span> <span m="5093510">do</span> <span m="5093670">that</span> <span m="5093850">usually</span> <span m="5094350">is</span> <span m="5094510">to</span> <span m="5094560">look</span> <span m="5094750">at</span> <span m="5094800">this</span> <span m="5094970">equation</span> <span m="5096230">and</span> <span m="5096280">then</span> <span m="5096430">say,</span> <span m="5096910">oh,</span> <span m="5097120">wait</span> <span m="5097290">a</span> <span m="5097310">minute,</span> <span m="5098230">I'm</span> <span m="5098390">going</span> <span m="5098610">to</span> <span m="5098710">talk</span> <span m="5098990">about</span> <span m="5099310">sets</span> <span m="5099730">because</span> <span m="5100270">I</span> <span m="5100430">choose</span> <span m="5100800">n</span> <span m="5101080">out</span> <span m="5101260">of</span> <span m="5101370">3n</span> <span m="5101870">over</span> <span m="5102150">here.</span> <span m="5102860">So</span> <span m="5103220">out of</span> <span m="5103390">3n</span> <span m="5103790">elements.</span> <span m="5104940">And</span> <span m="5105540">maybe</span> <span m="5105760">I</span> <span m="5105800">can</span> <span m="5106010">divide</span> <span m="5106470">up</span> <span m="5107260">the</span> <span m="5107380">sets</span> <span m="5107810">in</span> <span m="5107890">some</span> <span m="5108110">kind</span> <span m="5108360">of</span> <span m="5108420">specific</span> <span m="5108950">choice.</span> <span m="5109780">So</span> <span m="5110080">tomorrow</span> <span m="5110650">during</span> <span m="5110840">recitation</span> <span m="5111470">you will</span> <span m="5111610">also</span> <span m="5111940">see</span> <span m="5112600">combinatorial</span> <span m="5113250">proof.</span> <span m="5115630">Thank</span> <span m="5116130">you.</span> </p>
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